LMFDB - Embedded newform 5054.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.19520000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 $ x^8 - 2x^7 - 12x^6 + 16x^5 + 50x^4 - 24x^3 - 72x^2 - 32*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.761439$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +2.26420 q^{10} +2.53402 q^{11} -1.40760 q^{12} +1.31420 q^{13} -1.00000 q^{14} +3.18710 q^{15} +1.00000 q^{16} -4.50663 q^{17} +1.01865 q^{18} -2.26420 q^{20} -1.40760 q^{21} -2.53402 q^{22} -7.70050 q^{23} +1.40760 q^{24} +0.126599 q^{25} -1.31420 q^{26} +5.65667 q^{27} +1.00000 q^{28} +1.79740 q^{29} -3.18710 q^{30} +2.25306 q^{31} -1.00000 q^{32} -3.56689 q^{33} +4.50663 q^{34} -2.26420 q^{35} -1.01865 q^{36} +2.02693 q^{37} -1.84987 q^{39} +2.26420 q^{40} -8.23088 q^{41} +1.40760 q^{42} +1.11121 q^{43} +2.53402 q^{44} +2.30643 q^{45} +7.70050 q^{46} +5.77571 q^{47} -1.40760 q^{48} +1.00000 q^{49} -0.126599 q^{50} +6.34355 q^{51} +1.31420 q^{52} +12.0832 q^{53} -5.65667 q^{54} -5.73752 q^{55} -1.00000 q^{56} -1.79740 q^{58} -5.02694 q^{59} +3.18710 q^{60} -9.44679 q^{61} -2.25306 q^{62} -1.01865 q^{63} +1.00000 q^{64} -2.97560 q^{65} +3.56689 q^{66} +1.10809 q^{67} -4.50663 q^{68} +10.8393 q^{69} +2.26420 q^{70} +11.7694 q^{71} +1.01865 q^{72} -10.2135 q^{73} -2.02693 q^{74} -0.178202 q^{75} +2.53402 q^{77} +1.84987 q^{78} -2.75261 q^{79} -2.26420 q^{80} -4.90640 q^{81} +8.23088 q^{82} -11.7882 q^{83} -1.40760 q^{84} +10.2039 q^{85} -1.11121 q^{86} -2.53002 q^{87} -2.53402 q^{88} -13.4337 q^{89} -2.30643 q^{90} +1.31420 q^{91} -7.70050 q^{92} -3.17142 q^{93} -5.77571 q^{94} +1.40760 q^{96} +0.637330 q^{97} -1.00000 q^{98} -2.58128 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 2 * q^10 - 12 * q^11 + 4 * q^12 + 10 * q^13 - 8 * q^14 + 24 * q^15 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 + 22 * q^27 + 8 * q^28 + 8 * q^29 - 24 * q^30 + 18 * q^31 - 8 * q^32 - 16 * q^33 + 6 * q^34 - 2 * q^35 + 8 * q^36 + 36 * q^37 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 + 20 * q^46 - 10 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 + 12 * q^51 + 10 * q^52 + 32 * q^53 - 22 * q^54 - 22 * q^55 - 8 * q^56 - 8 * q^58 - 2 * q^59 + 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 + 44 * q^67 - 6 * q^68 + 2 * q^70 + 8 * q^71 - 8 * q^72 + 30 * q^73 - 36 * q^74 - 16 * q^75 - 12 * q^77 + 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 + 4 * q^84 - 16 * q^85 + 16 * q^86 + 24 * q^87 + 12 * q^88 - 22 * q^89 - 8 * q^90 + 10 * q^91 - 20 * q^92 - 16 * q^93 + 10 * q^94 - 4 * q^96 - 6 * q^97 - 8 * q^98 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.40760	−0.812681	−0.406340	−	0.913722i	$-0.633195\pi$
$3$	−1.40760	−0.812681	−0.406340	+	0.913722i	$0.633195\pi$
$4$	1.00000	0.500000
$5$	−2.26420	−1.01258	−0.506290	−	0.862363i	$-0.668984\pi$
$5$	−2.26420	−1.01258	−0.506290	+	0.862363i	$0.668984\pi$
$6$	1.40760	0.574652
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−1.01865	−0.339550
$10$	2.26420	0.716003
$11$	2.53402	0.764035	0.382018	−	0.924155i	$-0.375229\pi$
$11$	2.53402	0.764035	0.382018	+	0.924155i	$0.375229\pi$
$12$	−1.40760	−0.406340
$13$	1.31420	0.364492	0.182246	−	0.983253i	$-0.441663\pi$
$13$	1.31420	0.364492	0.182246	+	0.983253i	$0.441663\pi$
$14$	−1.00000	−0.267261
$15$	3.18710	0.822905
$16$	1.00000	0.250000
$17$	−4.50663	−1.09302	−0.546509	−	0.837453i	$-0.684044\pi$
$17$	−4.50663	−1.09302	−0.546509	+	0.837453i	$0.684044\pi$
$18$	1.01865	0.240098
$19$	0	0
$19$	0	0
$20$	−2.26420	−0.506290
$21$	−1.40760	−0.307164
$22$	−2.53402	−0.540254
$23$	−7.70050	−1.60566	−0.802832	−	0.596205i	$-0.796675\pi$
$23$	−7.70050	−1.60566	−0.802832	+	0.596205i	$0.796675\pi$
$24$	1.40760	0.287326
$25$	0.126599	0.0253199
$26$	−1.31420	−0.257735
$27$	5.65667	1.08863
$28$	1.00000	0.188982
$29$	1.79740	0.333768	0.166884	−	0.985977i	$-0.446629\pi$
$29$	1.79740	0.333768	0.166884	+	0.985977i	$0.446629\pi$
$30$	−3.18710	−0.581882
$31$	2.25306	0.404661	0.202331	−	0.979317i	$-0.435148\pi$
$31$	2.25306	0.404661	0.202331	+	0.979317i	$0.435148\pi$
$32$	−1.00000	−0.176777
$33$	−3.56689	−0.620917
$34$	4.50663	0.772881
$35$	−2.26420	−0.382720
$36$	−1.01865	−0.169775
$37$	2.02693	0.333225	0.166612	−	0.986022i	$-0.446717\pi$
$37$	2.02693	0.333225	0.166612	+	0.986022i	$0.446717\pi$
$38$	0	0
$39$	−1.84987	−0.296216
$40$	2.26420	0.358001
$41$	−8.23088	−1.28545	−0.642724	−	0.766098i	$-0.722196\pi$
$41$	−8.23088	−1.28545	−0.642724	+	0.766098i	$0.722196\pi$
$42$	1.40760	0.217198
$43$	1.11121	0.169459	0.0847293	−	0.996404i	$-0.472997\pi$
$43$	1.11121	0.169459	0.0847293	+	0.996404i	$0.472997\pi$
$44$	2.53402	0.382018
$45$	2.30643	0.343822
$46$	7.70050	1.13538
$47$	5.77571	0.842474	0.421237	−	0.906951i	$-0.361596\pi$
$47$	5.77571	0.842474	0.421237	+	0.906951i	$0.361596\pi$
$48$	−1.40760	−0.203170
$49$	1.00000	0.142857
$50$	−0.126599	−0.0179039
$51$	6.34355	0.888275
$52$	1.31420	0.182246
$53$	12.0832	1.65975	0.829876	−	0.557948i	$-0.188411\pi$
$53$	12.0832	1.65975	0.829876	+	0.557948i	$0.188411\pi$
$54$	−5.65667	−0.769775
$55$	−5.73752	−0.773647
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−1.79740	−0.236010
$59$	−5.02694	−0.654452	−0.327226	−	0.944946i	$-0.606114\pi$
$59$	−5.02694	−0.654452	−0.327226	+	0.944946i	$0.606114\pi$
$60$	3.18710	0.411452
$61$	−9.44679	−1.20954	−0.604769	−	0.796401i	$-0.706734\pi$
$61$	−9.44679	−1.20954	−0.604769	+	0.796401i	$0.706734\pi$
$62$	−2.25306	−0.286139
$63$	−1.01865	−0.128338
$64$	1.00000	0.125000
$65$	−2.97560	−0.369078
$66$	3.56689	0.439054
$67$	1.10809	0.135375	0.0676876	−	0.997707i	$-0.478438\pi$
$67$	1.10809	0.135375	0.0676876	+	0.997707i	$0.478438\pi$
$68$	−4.50663	−0.546509
$69$	10.8393	1.30489
$70$	2.26420	0.270624
$71$	11.7694	1.39678	0.698388	−	0.715720i	$-0.253901\pi$
$71$	11.7694	1.39678	0.698388	+	0.715720i	$0.253901\pi$
$72$	1.01865	0.120049
$73$	−10.2135	−1.19540	−0.597699	−	0.801720i	$-0.703918\pi$
$73$	−10.2135	−1.19540	−0.597699	+	0.801720i	$0.703918\pi$
$74$	−2.02693	−0.235625
$75$	−0.178202	−0.0205770
$76$	0	0
$77$	2.53402	0.288778
$78$	1.84987	0.209456
$79$	−2.75261	−0.309693	−0.154847	−	0.987939i	$-0.549488\pi$
$79$	−2.75261	−0.309693	−0.154847	+	0.987939i	$0.549488\pi$
$80$	−2.26420	−0.253145
$81$	−4.90640	−0.545156
$82$	8.23088	0.908949
$83$	−11.7882	−1.29392	−0.646962	−	0.762522i	$-0.723961\pi$
$83$	−11.7882	−1.29392	−0.646962	+	0.762522i	$0.723961\pi$
$84$	−1.40760	−0.153582
$85$	10.2039	1.10677
$86$	−1.11121	−0.119825
$87$	−2.53002	−0.271247
$88$	−2.53402	−0.270127
$89$	−13.4337	−1.42397	−0.711986	−	0.702194i	$-0.752204\pi$
$89$	−13.4337	−1.42397	−0.711986	+	0.702194i	$0.752204\pi$
$90$	−2.30643	−0.243119
$91$	1.31420	0.137765
$92$	−7.70050	−0.802832
$93$	−3.17142	−0.328861
$94$	−5.77571	−0.595719
$95$	0	0
$96$	1.40760	0.143663
$97$	0.637330	0.0647111	0.0323555	−	0.999476i	$-0.489699\pi$
$97$	0.637330	0.0647111	0.0323555	+	0.999476i	$0.489699\pi$
$98$	−1.00000	−0.101015
$99$	−2.58128	−0.259428
$100$	0.126599	0.0126599
$101$	−3.11677	−0.310130	−0.155065	−	0.987904i	$-0.549559\pi$
$101$	−3.11677	−0.310130	−0.155065	+	0.987904i	$0.549559\pi$
$102$	−6.34355	−0.628105
$103$	1.21127	0.119350	0.0596750	−	0.998218i	$-0.480994\pi$
$103$	1.21127	0.119350	0.0596750	+	0.998218i	$0.480994\pi$
$104$	−1.31420	−0.128867
$105$	3.18710	0.311029
$106$	−12.0832	−1.17362
$107$	−19.9061	−1.92439	−0.962195	−	0.272361i	$-0.912196\pi$
$107$	−19.9061	−1.92439	−0.962195	+	0.272361i	$0.912196\pi$
$108$	5.65667	0.544313
$109$	5.76507	0.552193	0.276097	−	0.961130i	$-0.410959\pi$
$109$	5.76507	0.552193	0.276097	+	0.961130i	$0.410959\pi$
$110$	5.73752	0.547051
$111$	−2.85311	−0.270805
$112$	1.00000	0.0944911
$113$	−10.7913	−1.01516	−0.507580	−	0.861604i	$-0.669460\pi$
$113$	−10.7913	−1.01516	−0.507580	+	0.861604i	$0.669460\pi$
$114$	0	0
$115$	17.4355	1.62587
$116$	1.79740	0.166884
$117$	−1.33871	−0.123763
$118$	5.02694	0.462767
$119$	−4.50663	−0.413122
$120$	−3.18710	−0.290941
$121$	−4.57875	−0.416250
$122$	9.44679	0.855272
$123$	11.5858	1.04466
$124$	2.25306	0.202331
$125$	11.0344	0.986942
$126$	1.01865	0.0907486
$127$	11.4309	1.01433	0.507165	−	0.861849i	$-0.330693\pi$
$127$	11.4309	1.01433	0.507165	+	0.861849i	$0.330693\pi$
$128$	−1.00000	−0.0883883
$129$	−1.56415	−0.137716
$130$	2.97560	0.260978
$131$	19.7348	1.72424	0.862121	−	0.506703i	$-0.169136\pi$
$131$	19.7348	1.72424	0.862121	+	0.506703i	$0.169136\pi$
$132$	−3.56689	−0.310458
$133$	0	0
$134$	−1.10809	−0.0957248
$135$	−12.8078	−1.10232
$136$	4.50663	0.386440
$137$	−17.4210	−1.48838	−0.744189	−	0.667969i	$-0.767164\pi$
$137$	−17.4210	−1.48838	−0.744189	+	0.667969i	$0.767164\pi$
$138$	−10.8393	−0.922698
$139$	6.38450	0.541526	0.270763	−	0.962646i	$-0.412724\pi$
$139$	6.38450	0.541526	0.270763	+	0.962646i	$0.412724\pi$
$140$	−2.26420	−0.191360
$141$	−8.12991	−0.684662
$142$	−11.7694	−0.987669
$143$	3.33020	0.278485
$144$	−1.01865	−0.0848875
$145$	−4.06967	−0.337967
$146$	10.2135	0.845275
$147$	−1.40760	−0.116097
$148$	2.02693	0.166612
$149$	−18.3360	−1.50215	−0.751073	−	0.660219i	$-0.770463\pi$
$149$	−18.3360	−1.50215	−0.751073	+	0.660219i	$0.770463\pi$
$150$	0.178202	0.0145501
$151$	17.2670	1.40517	0.702586	−	0.711599i	$-0.252028\pi$
$151$	17.2670	1.40517	0.702586	+	0.711599i	$0.252028\pi$
$152$	0	0
$153$	4.59068	0.371135
$154$	−2.53402	−0.204197
$155$	−5.10138	−0.409752
$156$	−1.84987	−0.148108
$157$	7.87418	0.628428	0.314214	−	0.949352i	$-0.398259\pi$
$157$	7.87418	0.628428	0.314214	+	0.949352i	$0.398259\pi$
$158$	2.75261	0.218986
$159$	−17.0083	−1.34885
$160$	2.26420	0.179001
$161$	−7.70050	−0.606884
$162$	4.90640	0.385483
$163$	0.922928	0.0722893	0.0361447	−	0.999347i	$-0.488492\pi$
$163$	0.922928	0.0722893	0.0361447	+	0.999347i	$0.488492\pi$
$164$	−8.23088	−0.642724
$165$	8.07616	0.628728
$166$	11.7882	0.914943
$167$	−11.4713	−0.887678	−0.443839	−	0.896107i	$-0.646384\pi$
$167$	−11.4713	−0.887678	−0.443839	+	0.896107i	$0.646384\pi$
$168$	1.40760	0.108599
$169$	−11.2729	−0.867145
$170$	−10.2039	−0.782604
$171$	0	0
$172$	1.11121	0.0847293
$173$	1.01941	0.0775042	0.0387521	−	0.999249i	$-0.487662\pi$
$173$	1.01941	0.0775042	0.0387521	+	0.999249i	$0.487662\pi$
$174$	2.53002	0.191801
$175$	0.126599	0.00957002
$176$	2.53402	0.191009
$177$	7.07594	0.531860
$178$	13.4337	1.00690
$179$	6.16884	0.461080	0.230540	−	0.973063i	$-0.425951\pi$
$179$	6.16884	0.461080	0.230540	+	0.973063i	$0.425951\pi$
$180$	2.30643	0.171911
$181$	17.9933	1.33743	0.668716	−	0.743518i	$-0.266844\pi$
$181$	17.9933	1.33743	0.668716	+	0.743518i	$0.266844\pi$
$182$	−1.31420	−0.0974147
$183$	13.2973	0.982968
$184$	7.70050	0.567688
$185$	−4.58936	−0.337417
$186$	3.17142	0.232540
$187$	−11.4199	−0.835105
$188$	5.77571	0.421237
$189$	5.65667	0.411462
$190$	0	0
$191$	3.40620	0.246464	0.123232	−	0.992378i	$-0.460674\pi$
$191$	3.40620	0.246464	0.123232	+	0.992378i	$0.460674\pi$
$192$	−1.40760	−0.101585
$193$	3.90426	0.281035	0.140517	−	0.990078i	$-0.455123\pi$
$193$	3.90426	0.281035	0.140517	+	0.990078i	$0.455123\pi$
$194$	−0.637330	−0.0457577
$195$	4.18847	0.299943
$196$	1.00000	0.0714286
$197$	−2.65223	−0.188963	−0.0944817	−	0.995527i	$-0.530119\pi$
$197$	−2.65223	−0.188963	−0.0944817	+	0.995527i	$0.530119\pi$
$198$	2.58128	0.183443
$199$	12.4354	0.881520	0.440760	−	0.897625i	$-0.354709\pi$
$199$	12.4354	0.881520	0.440760	+	0.897625i	$0.354709\pi$
$200$	−0.126599	−0.00895193
$201$	−1.55976	−0.110017
$202$	3.11677	0.219295
$203$	1.79740	0.126153
$204$	6.34355	0.444138
$205$	18.6364	1.30162
$206$	−1.21127	−0.0843932
$207$	7.84411	0.545204
$208$	1.31420	0.0911231
$209$	0	0
$210$	−3.18710	−0.219931
$211$	14.5888	1.00433	0.502166	−	0.864771i	$-0.332537\pi$
$211$	14.5888	1.00433	0.502166	+	0.864771i	$0.332537\pi$
$212$	12.0832	0.829876
$213$	−16.5667	−1.13513
$214$	19.9061	1.36075
$215$	−2.51601	−0.171591
$216$	−5.65667	−0.384888
$217$	2.25306	0.152948
$218$	−5.76507	−0.390460
$219$	14.3766	0.971477
$220$	−5.73752	−0.386824
$221$	−5.92260	−0.398397
$222$	2.85311	0.191488
$223$	−16.8285	−1.12692	−0.563459	−	0.826144i	$-0.690530\pi$
$223$	−16.8285	−1.12692	−0.563459	+	0.826144i	$0.690530\pi$
$224$	−1.00000	−0.0668153
$225$	−0.128961	−0.00859737
$226$	10.7913	0.717827
$227$	−4.45322	−0.295570	−0.147785	−	0.989019i	$-0.547214\pi$
$227$	−4.45322	−0.295570	−0.147785	+	0.989019i	$0.547214\pi$
$228$	0	0
$229$	20.1895	1.33416	0.667081	−	0.744985i	$-0.267544\pi$
$229$	20.1895	1.33416	0.667081	+	0.744985i	$0.267544\pi$
$230$	−17.4355	−1.14966
$231$	−3.56689	−0.234684
$232$	−1.79740	−0.118005
$233$	20.3141	1.33082	0.665410	−	0.746478i	$-0.268257\pi$
$233$	20.3141	1.33082	0.665410	+	0.746478i	$0.268257\pi$
$234$	1.33871	0.0875139
$235$	−13.0774	−0.853073
$236$	−5.02694	−0.327226
$237$	3.87459	0.251682
$238$	4.50663	0.292122
$239$	25.8369	1.67125	0.835625	−	0.549300i	$-0.185105\pi$
$239$	25.8369	1.67125	0.835625	+	0.549300i	$0.185105\pi$
$240$	3.18710	0.205726
$241$	16.3177	1.05112	0.525558	−	0.850757i	$-0.323856\pi$
$241$	16.3177	1.05112	0.525558	+	0.850757i	$0.323856\pi$
$242$	4.57875	0.294333
$243$	−10.0637	−0.645589
$244$	−9.44679	−0.604769
$245$	−2.26420	−0.144654
$246$	−11.5858	−0.738685
$247$	0	0
$248$	−2.25306	−0.143069
$249$	16.5931	1.05155
$250$	−11.0344	−0.697874
$251$	−13.4191	−0.847005	−0.423502	−	0.905895i	$-0.639199\pi$
$251$	−13.4191	−0.847005	−0.423502	+	0.905895i	$0.639199\pi$
$252$	−1.01865	−0.0641689
$253$	−19.5132	−1.22678
$254$	−11.4309	−0.717240
$255$	−14.3631	−0.899450
$256$	1.00000	0.0625000
$257$	6.56928	0.409780	0.204890	−	0.978785i	$-0.434316\pi$
$257$	6.56928	0.409780	0.204890	+	0.978785i	$0.434316\pi$
$258$	1.56415	0.0973797
$259$	2.02693	0.125947
$260$	−2.97560	−0.184539
$261$	−1.83092	−0.113331
$262$	−19.7348	−1.21922
$263$	28.6089	1.76410	0.882051	−	0.471154i	$-0.156163\pi$
$263$	28.6089	1.76410	0.882051	+	0.471154i	$0.156163\pi$
$264$	3.56689	0.219527
$265$	−27.3587	−1.68063
$266$	0	0
$267$	18.9094	1.15723
$268$	1.10809	0.0676876
$269$	19.7174	1.20219	0.601095	−	0.799178i	$-0.294731\pi$
$269$	19.7174	1.20219	0.601095	+	0.799178i	$0.294731\pi$
$270$	12.8078	0.779460
$271$	6.93759	0.421429	0.210714	−	0.977548i	$-0.432421\pi$
$271$	6.93759	0.421429	0.210714	+	0.977548i	$0.432421\pi$
$272$	−4.50663	−0.273255
$273$	−1.84987	−0.111959
$274$	17.4210	1.05244
$275$	0.320805	0.0193453
$276$	10.8393	0.652446
$277$	−22.1848	−1.33296	−0.666478	−	0.745525i	$-0.732199\pi$
$277$	−22.1848	−1.33296	−0.666478	+	0.745525i	$0.732199\pi$
$278$	−6.38450	−0.382917
$279$	−2.29508	−0.137403
$280$	2.26420	0.135312
$281$	32.8994	1.96262	0.981308	−	0.192443i	$-0.0616410\pi$
$281$	32.8994	1.96262	0.981308	+	0.192443i	$0.0616410\pi$
$282$	8.12991	0.484129
$283$	−2.48145	−0.147507	−0.0737533	−	0.997277i	$-0.523498\pi$
$283$	−2.48145	−0.147507	−0.0737533	+	0.997277i	$0.523498\pi$
$284$	11.7694	0.698388
$285$	0	0
$286$	−3.33020	−0.196919
$287$	−8.23088	−0.485854
$288$	1.01865	0.0600245
$289$	3.30973	0.194690
$290$	4.06967	0.238979
$291$	−0.897109	−0.0525895
$292$	−10.2135	−0.597699
$293$	0.143516	0.00838429	0.00419214	−	0.999991i	$-0.498666\pi$
$293$	0.143516	0.00838429	0.00419214	+	0.999991i	$0.498666\pi$
$294$	1.40760	0.0820931
$295$	11.3820	0.662685
$296$	−2.02693	−0.117813
$297$	14.3341	0.831749
$298$	18.3360	1.06218
$299$	−10.1200	−0.585252
$300$	−0.178202	−0.0102885
$301$	1.11121	0.0640493
$302$	−17.2670	−0.993607
$303$	4.38717	0.252036
$304$	0	0
$305$	21.3894	1.22475
$306$	−4.59068	−0.262432
$307$	23.3704	1.33382	0.666910	−	0.745138i	$-0.267616\pi$
$307$	23.3704	1.33382	0.666910	+	0.745138i	$0.267616\pi$
$308$	2.53402	0.144389
$309$	−1.70499	−0.0969934
$310$	5.10138	0.289739
$311$	−16.8078	−0.953086	−0.476543	−	0.879151i	$-0.658110\pi$
$311$	−16.8078	−0.953086	−0.476543	+	0.879151i	$0.658110\pi$
$312$	1.84987	0.104728
$313$	14.1649	0.800650	0.400325	−	0.916373i	$-0.368897\pi$
$313$	14.1649	0.800650	0.400325	+	0.916373i	$0.368897\pi$
$314$	−7.87418	−0.444366
$315$	2.30643	0.129952
$316$	−2.75261	−0.154847
$317$	−15.7154	−0.882662	−0.441331	−	0.897344i	$-0.645494\pi$
$317$	−15.7154	−0.882662	−0.441331	+	0.897344i	$0.645494\pi$
$318$	17.0083	0.953780
$319$	4.55464	0.255011
$320$	−2.26420	−0.126573
$321$	28.0198	1.56392
$322$	7.70050	0.429132
$323$	0	0
$324$	−4.90640	−0.272578
$325$	0.166376	0.00922891
$326$	−0.922928	−0.0511163
$327$	−8.11493	−0.448757
$328$	8.23088	0.454474
$329$	5.77571	0.318425
$330$	−8.07616	−0.444578
$331$	−7.84789	−0.431359	−0.215680	−	0.976464i	$-0.569197\pi$
$331$	−7.84789	−0.431359	−0.215680	+	0.976464i	$0.569197\pi$
$332$	−11.7882	−0.646962
$333$	−2.06473	−0.113146
$334$	11.4713	0.627683
$335$	−2.50895	−0.137078
$336$	−1.40760	−0.0767911
$337$	32.7210	1.78243	0.891213	−	0.453584i	$-0.149855\pi$
$337$	32.7210	1.78243	0.891213	+	0.453584i	$0.149855\pi$
$338$	11.2729	0.613164
$339$	15.1899	0.825002
$340$	10.2039	0.553385
$341$	5.70929	0.309176
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.11121	−0.0599127
$345$	−24.5422	−1.32131
$346$	−1.01941	−0.0548038
$347$	33.6602	1.80697	0.903486	−	0.428617i	$-0.140999\pi$
$347$	33.6602	1.80697	0.903486	+	0.428617i	$0.140999\pi$
$348$	−2.53002	−0.135624
$349$	−2.56580	−0.137344	−0.0686721	−	0.997639i	$-0.521876\pi$
$349$	−2.56580	−0.137344	−0.0686721	+	0.997639i	$0.521876\pi$
$350$	−0.126599	−0.00676703
$351$	7.43397	0.396796
$352$	−2.53402	−0.135064
$353$	−22.9673	−1.22243	−0.611213	−	0.791466i	$-0.709318\pi$
$353$	−22.9673	−1.22243	−0.611213	+	0.791466i	$0.709318\pi$
$354$	−7.07594	−0.376082
$355$	−26.6484	−1.41435
$356$	−13.4337	−0.711986
$357$	6.34355	0.335736
$358$	−6.16884	−0.326033
$359$	−5.36264	−0.283029	−0.141515	−	0.989936i	$-0.545197\pi$
$359$	−5.36264	−0.283029	−0.141515	+	0.989936i	$0.545197\pi$
$360$	−2.30643	−0.121559
$361$	0	0
$362$	−17.9933	−0.945708
$363$	6.44507	0.338279
$364$	1.31420	0.0688826
$365$	23.1254	1.21044
$366$	−13.2973	−0.695063
$367$	−22.4786	−1.17337	−0.586687	−	0.809814i	$-0.699568\pi$
$367$	−22.4786	−1.17337	−0.586687	+	0.809814i	$0.699568\pi$
$368$	−7.70050	−0.401416
$369$	8.38439	0.436474
$370$	4.58936	0.238590
$371$	12.0832	0.627327
$372$	−3.17142	−0.164430
$373$	−12.3028	−0.637015	−0.318507	−	0.947920i	$-0.603182\pi$
$373$	−12.3028	−0.637015	−0.318507	+	0.947920i	$0.603182\pi$
$374$	11.4199	0.590508
$375$	−15.5320	−0.802069
$376$	−5.77571	−0.297859
$377$	2.36213	0.121656
$378$	−5.65667	−0.290948
$379$	−9.72581	−0.499581	−0.249791	−	0.968300i	$-0.580362\pi$
$379$	−9.72581	−0.499581	−0.249791	+	0.968300i	$0.580362\pi$
$380$	0	0
$381$	−16.0902	−0.824327
$382$	−3.40620	−0.174277
$383$	−15.4098	−0.787402	−0.393701	−	0.919239i	$-0.628806\pi$
$383$	−15.4098	−0.787402	−0.393701	+	0.919239i	$0.628806\pi$
$384$	1.40760	0.0718315
$385$	−5.73752	−0.292411
$386$	−3.90426	−0.198722
$387$	−1.13194	−0.0575397
$388$	0.637330	0.0323555
$389$	2.58927	0.131281	0.0656405	−	0.997843i	$-0.479091\pi$
$389$	2.58927	0.131281	0.0656405	+	0.997843i	$0.479091\pi$
$390$	−4.18847	−0.212091
$391$	34.7033	1.75502
$392$	−1.00000	−0.0505076
$393$	−27.7789	−1.40126
$394$	2.65223	0.133617
$395$	6.23246	0.313589
$396$	−2.58128	−0.129714
$397$	11.2929	0.566772	0.283386	−	0.959006i	$-0.408542\pi$
$397$	11.2929	0.566772	0.283386	+	0.959006i	$0.408542\pi$
$398$	−12.4354	−0.623329
$399$	0	0
$400$	0.126599	0.00632997
$401$	19.0880	0.953210	0.476605	−	0.879118i	$-0.341867\pi$
$401$	19.0880	0.953210	0.476605	+	0.879118i	$0.341867\pi$
$402$	1.55976	0.0777937
$403$	2.96096	0.147496
$404$	−3.11677	−0.155065
$405$	11.1091	0.552014
$406$	−1.79740	−0.0892033
$407$	5.13627	0.254595
$408$	−6.34355	−0.314053
$409$	30.7422	1.52010	0.760052	−	0.649862i	$-0.225173\pi$
$409$	30.7422	1.52010	0.760052	+	0.649862i	$0.225173\pi$
$410$	−18.6364	−0.920384
$411$	24.5219	1.20958
$412$	1.21127	0.0596750
$413$	−5.02694	−0.247359
$414$	−7.84411	−0.385517
$415$	26.6909	1.31020
$416$	−1.31420	−0.0644337
$417$	−8.98685	−0.440088
$418$	0	0
$419$	−24.3506	−1.18961	−0.594803	−	0.803872i	$-0.702770\pi$
$419$	−24.3506	−1.18961	−0.594803	+	0.803872i	$0.702770\pi$
$420$	3.18710	0.155514
$421$	5.28504	0.257577	0.128789	−	0.991672i	$-0.458891\pi$
$421$	5.28504	0.257577	0.128789	+	0.991672i	$0.458891\pi$
$422$	−14.5888	−0.710170
$423$	−5.88343	−0.286062
$424$	−12.0832	−0.586811
$425$	−0.570537	−0.0276751
$426$	16.5667	0.802660
$427$	−9.44679	−0.457162
$428$	−19.9061	−0.962195
$429$	−4.68760	−0.226319
$430$	2.51601	0.121333
$431$	−0.618814	−0.0298072	−0.0149036	−	0.999889i	$-0.504744\pi$
$431$	−0.618814	−0.0298072	−0.0149036	+	0.999889i	$0.504744\pi$
$432$	5.65667	0.272157
$433$	−36.8806	−1.77237	−0.886185	−	0.463331i	$-0.846654\pi$
$433$	−36.8806	−1.77237	−0.886185	+	0.463331i	$0.846654\pi$
$434$	−2.25306	−0.108150
$435$	5.72848	0.274660
$436$	5.76507	0.276097
$437$	0	0
$438$	−14.3766	−0.686938
$439$	−39.8288	−1.90092	−0.950462	−	0.310840i	$-0.899390\pi$
$439$	−39.8288	−1.90092	−0.950462	+	0.310840i	$0.899390\pi$
$440$	5.73752	0.273526
$441$	−1.01865	−0.0485072
$442$	5.92260	0.281709
$443$	37.2011	1.76748	0.883739	−	0.467980i	$-0.155018\pi$
$443$	37.2011	1.76748	0.883739	+	0.467980i	$0.155018\pi$
$444$	−2.85311	−0.135403
$445$	30.4166	1.44189
$446$	16.8285	0.796852
$447$	25.8099	1.22077
$448$	1.00000	0.0472456
$449$	27.2063	1.28395	0.641973	−	0.766728i	$-0.278116\pi$
$449$	27.2063	1.28395	0.641973	+	0.766728i	$0.278116\pi$
$450$	0.128961	0.00607926
$451$	−20.8572	−0.982127
$452$	−10.7913	−0.507580
$453$	−24.3052	−1.14196
$454$	4.45322	0.209000
$455$	−2.97560	−0.139498
$456$	0	0
$457$	5.70914	0.267062	0.133531	−	0.991045i	$-0.457368\pi$
$457$	5.70914	0.267062	0.133531	+	0.991045i	$0.457368\pi$
$458$	−20.1895	−0.943395
$459$	−25.4925	−1.18989
$460$	17.4355	0.812933
$461$	18.4972	0.861499	0.430749	−	0.902472i	$-0.358249\pi$
$461$	18.4972	0.861499	0.430749	+	0.902472i	$0.358249\pi$
$462$	3.56689	0.165947
$463$	29.8307	1.38635	0.693175	−	0.720769i	$-0.256211\pi$
$463$	29.8307	1.38635	0.693175	+	0.720769i	$0.256211\pi$
$464$	1.79740	0.0834421
$465$	7.18072	0.332998
$466$	−20.3141	−0.941032
$467$	34.9102	1.61545	0.807726	−	0.589559i	$-0.200698\pi$
$467$	34.9102	1.61545	0.807726	+	0.589559i	$0.200698\pi$
$468$	−1.33871	−0.0618817
$469$	1.10809	0.0511670
$470$	13.0774	0.603213
$471$	−11.0837	−0.510711
$472$	5.02694	0.231384
$473$	2.81584	0.129472
$474$	−3.87459	−0.177966
$475$	0	0
$476$	−4.50663	−0.206561
$477$	−12.3085	−0.563569
$478$	−25.8369	−1.18175
$479$	1.48027	0.0676354	0.0338177	−	0.999428i	$-0.489233\pi$
$479$	1.48027	0.0676354	0.0338177	+	0.999428i	$0.489233\pi$
$480$	−3.18710	−0.145470
$481$	2.66378	0.121458
$482$	−16.3177	−0.743252
$483$	10.8393	0.493203
$484$	−4.57875	−0.208125
$485$	−1.44304	−0.0655252
$486$	10.0637	0.456500
$487$	29.1423	1.32057	0.660283	−	0.751017i	$-0.270436\pi$
$487$	29.1423	1.32057	0.660283	+	0.751017i	$0.270436\pi$
$488$	9.44679	0.427636
$489$	−1.29912	−0.0587481
$490$	2.26420	0.102286
$491$	−7.07800	−0.319426	−0.159713	−	0.987164i	$-0.551057\pi$
$491$	−7.07800	−0.319426	−0.159713	+	0.987164i	$0.551057\pi$
$492$	11.5858	0.522329
$493$	−8.10021	−0.364815
$494$	0	0
$495$	5.84453	0.262692
$496$	2.25306	0.101165
$497$	11.7694	0.527931
$498$	−16.5931	−0.743557
$499$	11.0798	0.496000	0.248000	−	0.968760i	$-0.420227\pi$
$499$	11.0798	0.496000	0.248000	+	0.968760i	$0.420227\pi$
$500$	11.0344	0.493471
$501$	16.1471	0.721399
$502$	13.4191	0.598923
$503$	23.3118	1.03942	0.519711	−	0.854342i	$-0.326040\pi$
$503$	23.3118	1.03942	0.519711	+	0.854342i	$0.326040\pi$
$504$	1.01865	0.0453743
$505$	7.05698	0.314031
$506$	19.5132	0.867467
$507$	15.8678	0.704712
$508$	11.4309	0.507165
$509$	25.2949	1.12118	0.560588	−	0.828095i	$-0.310575\pi$
$509$	25.2949	1.12118	0.560588	+	0.828095i	$0.310575\pi$
$510$	14.3631	0.636007
$511$	−10.2135	−0.451818
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−6.56928	−0.289758
$515$	−2.74256	−0.120852
$516$	−1.56415	−0.0688579
$517$	14.6357	0.643679
$518$	−2.02693	−0.0890580
$519$	−1.43492	−0.0629862
$520$	2.97560	0.130489
$521$	−5.59439	−0.245095	−0.122547	−	0.992463i	$-0.539106\pi$
$521$	−5.59439	−0.245095	−0.122547	+	0.992463i	$0.539106\pi$
$522$	1.83092	0.0801372
$523$	−38.7757	−1.69554	−0.847771	−	0.530362i	$-0.822056\pi$
$523$	−38.7757	−1.69554	−0.847771	+	0.530362i	$0.822056\pi$
$524$	19.7348	0.862121
$525$	−0.178202	−0.00777737
$526$	−28.6089	−1.24741
$527$	−10.1537	−0.442302
$528$	−3.56689	−0.155229
$529$	36.2977	1.57816
$530$	27.3587	1.18839
$531$	5.12069	0.222219
$532$	0	0
$533$	−10.8170	−0.468536
$534$	−18.9094	−0.818288
$535$	45.0713	1.94860
$536$	−1.10809	−0.0478624
$537$	−8.68328	−0.374711
$538$	−19.7174	−0.850077
$539$	2.53402	0.109148
$540$	−12.8078	−0.551161
$541$	18.8155	0.808943	0.404472	−	0.914551i	$-0.367455\pi$
$541$	18.8155	0.808943	0.404472	+	0.914551i	$0.367455\pi$
$542$	−6.93759	−0.297995
$543$	−25.3275	−1.08691
$544$	4.50663	0.193220
$545$	−13.0533	−0.559140
$546$	1.84987	0.0791670
$547$	5.85379	0.250290	0.125145	−	0.992138i	$-0.460060\pi$
$547$	5.85379	0.250290	0.125145	+	0.992138i	$0.460060\pi$
$548$	−17.4210	−0.744189
$549$	9.62297	0.410699
$550$	−0.320805	−0.0136792
$551$	0	0
$552$	−10.8393	−0.461349
$553$	−2.75261	−0.117053
$554$	22.1848	0.942542
$555$	6.46001	0.274212
$556$	6.38450	0.270763
$557$	43.6742	1.85053	0.925267	−	0.379316i	$-0.123841\pi$
$557$	43.6742	1.85053	0.925267	+	0.379316i	$0.123841\pi$
$558$	2.29508	0.0971585
$559$	1.46035	0.0617664
$560$	−2.26420	−0.0956799
$561$	16.0747	0.678673
$562$	−32.8994	−1.38778
$563$	−3.05365	−0.128696	−0.0643479	−	0.997928i	$-0.520497\pi$
$563$	−3.05365	−0.128696	−0.0643479	+	0.997928i	$0.520497\pi$
$564$	−8.12991	−0.342331
$565$	24.4337	1.02793
$566$	2.48145	0.104303
$567$	−4.90640	−0.206049
$568$	−11.7694	−0.493835
$569$	33.6088	1.40895	0.704477	−	0.709726i	$-0.251181\pi$
$569$	33.6088	1.40895	0.704477	+	0.709726i	$0.251181\pi$
$570$	0	0
$571$	21.8909	0.916106	0.458053	−	0.888925i	$-0.348547\pi$
$571$	21.8909	0.916106	0.458053	+	0.888925i	$0.348547\pi$
$572$	3.33020	0.139242
$573$	−4.79459	−0.200297
$574$	8.23088	0.343550
$575$	−0.974879	−0.0406553
$576$	−1.01865	−0.0424438
$577$	41.7821	1.73941	0.869705	−	0.493571i	$-0.164309\pi$
$577$	41.7821	1.73941	0.869705	+	0.493571i	$0.164309\pi$
$578$	−3.30973	−0.137666
$579$	−5.49565	−0.228392
$580$	−4.06967	−0.168984
$581$	−11.7882	−0.489058
$582$	0.897109	0.0371864
$583$	30.6190	1.26811
$584$	10.2135	0.422637
$585$	3.03110	0.125320
$586$	−0.143516	−0.00592859
$587$	−32.6303	−1.34680	−0.673398	−	0.739280i	$-0.735166\pi$
$587$	−32.6303	−1.34680	−0.673398	+	0.739280i	$0.735166\pi$
$588$	−1.40760	−0.0580486
$589$	0	0
$590$	−11.3820	−0.468589
$591$	3.73329	0.153567
$592$	2.02693	0.0833061
$593$	0.644712	0.0264752	0.0132376	−	0.999912i	$-0.495786\pi$
$593$	0.644712	0.0264752	0.0132376	+	0.999912i	$0.495786\pi$
$594$	−14.3341	−0.588135
$595$	10.2039	0.418320
$596$	−18.3360	−0.751073
$597$	−17.5041	−0.716395
$598$	10.1200	0.413836
$599$	4.92246	0.201126	0.100563	−	0.994931i	$-0.467936\pi$
$599$	4.92246	0.201126	0.100563	+	0.994931i	$0.467936\pi$
$600$	0.178202	0.00727506
$601$	13.5325	0.552004	0.276002	−	0.961157i	$-0.410990\pi$
$601$	13.5325	0.552004	0.276002	+	0.961157i	$0.410990\pi$
$602$	−1.11121	−0.0452897
$603$	−1.12876	−0.0459667
$604$	17.2670	0.702586
$605$	10.3672	0.421487
$606$	−4.38717	−0.178217
$607$	7.47174	0.303269	0.151634	−	0.988437i	$-0.451546\pi$
$607$	7.47174	0.303269	0.151634	+	0.988437i	$0.451546\pi$
$608$	0	0
$609$	−2.53002	−0.102522
$610$	−21.3894	−0.866032
$611$	7.59041	0.307075
$612$	4.59068	0.185567
$613$	8.98220	0.362788	0.181394	−	0.983411i	$-0.441939\pi$
$613$	8.98220	0.362788	0.181394	+	0.983411i	$0.441939\pi$
$614$	−23.3704	−0.943153
$615$	−26.2326	−1.05780
$616$	−2.53402	−0.102098
$617$	4.62851	0.186337	0.0931683	−	0.995650i	$-0.470301\pi$
$617$	4.62851	0.186337	0.0931683	+	0.995650i	$0.470301\pi$
$618$	1.70499	0.0685847
$619$	−0.578478	−0.0232510	−0.0116255	−	0.999932i	$-0.503701\pi$
$619$	−0.578478	−0.0232510	−0.0116255	+	0.999932i	$0.503701\pi$
$620$	−5.10138	−0.204876
$621$	−43.5592	−1.74797
$622$	16.8078	0.673933
$623$	−13.4337	−0.538210
$624$	−1.84987	−0.0740540
$625$	−25.6170	−1.02468
$626$	−14.1649	−0.566145
$627$	0	0
$628$	7.87418	0.314214
$629$	−9.13461	−0.364221
$630$	−2.30643	−0.0918903
$631$	−44.3940	−1.76730	−0.883648	−	0.468152i	$-0.844920\pi$
$631$	−44.3940	−1.76730	−0.883648	+	0.468152i	$0.844920\pi$
$632$	2.75261	0.109493
$633$	−20.5352	−0.816201
$634$	15.7154	0.624136
$635$	−25.8819	−1.02709
$636$	−17.0083	−0.674424
$637$	1.31420	0.0520703
$638$	−4.55464	−0.180320
$639$	−11.9889	−0.474275
$640$	2.26420	0.0895003
$641$	−21.8734	−0.863948	−0.431974	−	0.901886i	$-0.642183\pi$
$641$	−21.8734	−0.863948	−0.431974	+	0.901886i	$0.642183\pi$
$642$	−28.0198	−1.10585
$643$	27.8528	1.09841	0.549204	−	0.835689i	$-0.314931\pi$
$643$	27.8528	1.09841	0.549204	+	0.835689i	$0.314931\pi$
$644$	−7.70050	−0.303442
$645$	3.54155	0.139448
$646$	0	0
$647$	−32.8926	−1.29314	−0.646570	−	0.762855i	$-0.723797\pi$
$647$	−32.8926	−1.29314	−0.646570	+	0.762855i	$0.723797\pi$
$648$	4.90640	0.192742
$649$	−12.7383	−0.500024
$650$	−0.166376	−0.00652582
$651$	−3.17142	−0.124298
$652$	0.922928	0.0361447
$653$	28.3623	1.10990	0.554951	−	0.831883i	$-0.312737\pi$
$653$	28.3623	1.10990	0.554951	+	0.831883i	$0.312737\pi$
$654$	8.11493	0.317319
$655$	−44.6836	−1.74593
$656$	−8.23088	−0.321362
$657$	10.4040	0.405898
$658$	−5.77571	−0.225161
$659$	−4.04510	−0.157575	−0.0787874	−	0.996891i	$-0.525105\pi$
$659$	−4.04510	−0.157575	−0.0787874	+	0.996891i	$0.525105\pi$
$660$	8.07616	0.314364
$661$	45.1752	1.75711	0.878556	−	0.477639i	$-0.158507\pi$
$661$	45.1752	1.75711	0.878556	+	0.477639i	$0.158507\pi$
$662$	7.84789	0.305017
$663$	8.33667	0.323769
$664$	11.7882	0.457472
$665$	0	0
$666$	2.06473	0.0800066
$667$	−13.8409	−0.535920
$668$	−11.4713	−0.443839
$669$	23.6878	0.915825
$670$	2.50895	0.0969291
$671$	−23.9383	−0.924129
$672$	1.40760	0.0542995
$673$	−24.4991	−0.944370	−0.472185	−	0.881500i	$-0.656534\pi$
$673$	−24.4991	−0.944370	−0.472185	+	0.881500i	$0.656534\pi$
$674$	−32.7210	−1.26037
$675$	0.716131	0.0275639
$676$	−11.2729	−0.433573
$677$	−51.1922	−1.96748	−0.983738	−	0.179610i	$-0.942516\pi$
$677$	−51.1922	−1.96748	−0.983738	+	0.179610i	$0.942516\pi$
$678$	−15.1899	−0.583364
$679$	0.637330	0.0244585
$680$	−10.2039	−0.391302
$681$	6.26837	0.240204
$682$	−5.70929	−0.218620
$683$	24.6139	0.941825	0.470912	−	0.882180i	$-0.343925\pi$
$683$	24.6139	0.941825	0.470912	+	0.882180i	$0.343925\pi$
$684$	0	0
$685$	39.4447	1.50710
$686$	−1.00000	−0.0381802
$687$	−28.4189	−1.08425
$688$	1.11121	0.0423646
$689$	15.8797	0.604967
$690$	24.5422	0.934307
$691$	27.6269	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$691$	27.6269	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$692$	1.01941	0.0387521
$693$	−2.58128	−0.0980546
$694$	−33.6602	−1.27772
$695$	−14.4558	−0.548339
$696$	2.53002	0.0959003
$697$	37.0935	1.40502
$698$	2.56580	0.0971170
$699$	−28.5942	−1.08153
$700$	0.126599	0.00478501
$701$	−26.1374	−0.987198	−0.493599	−	0.869690i	$-0.664319\pi$
$701$	−26.1374	−0.987198	−0.493599	+	0.869690i	$0.664319\pi$
$702$	−7.43397	−0.280577
$703$	0	0
$704$	2.53402	0.0955044
$705$	18.4077	0.693276
$706$	22.9673	0.864386
$707$	−3.11677	−0.117218
$708$	7.07594	0.265930
$709$	0.802841	0.0301513	0.0150757	−	0.999886i	$-0.495201\pi$
$709$	0.802841	0.0301513	0.0150757	+	0.999886i	$0.495201\pi$
$710$	26.6484	1.00009
$711$	2.80395	0.105156
$712$	13.4337	0.503450
$713$	−17.3497	−0.649750
$714$	−6.34355	−0.237402
$715$	−7.54023	−0.281988
$716$	6.16884	0.230540
$717$	−36.3681	−1.35819
$718$	5.36264	0.200132
$719$	−33.3899	−1.24523	−0.622616	−	0.782527i	$-0.713930\pi$
$719$	−33.3899	−1.24523	−0.622616	+	0.782527i	$0.713930\pi$
$720$	2.30643	0.0859555
$721$	1.21127	0.0451101
$722$	0	0
$723$	−22.9689	−0.854223
$724$	17.9933	0.668716
$725$	0.227550	0.00845098
$726$	−6.44507	−0.239199
$727$	−0.512039	−0.0189905	−0.00949524	−	0.999955i	$-0.503022\pi$
$727$	−0.512039	−0.0189905	−0.00949524	+	0.999955i	$0.503022\pi$
$728$	−1.31420	−0.0487073
$729$	28.8850	1.06981
$730$	−23.1254	−0.855909
$731$	−5.00783	−0.185221
$732$	13.2973	0.491484
$733$	0.898494	0.0331866	0.0165933	−	0.999862i	$-0.494718\pi$
$733$	0.898494	0.0331866	0.0165933	+	0.999862i	$0.494718\pi$
$734$	22.4786	0.829700
$735$	3.18710	0.117558
$736$	7.70050	0.283844
$737$	2.80793	0.103431
$738$	−8.38439	−0.308634
$739$	24.0173	0.883490	0.441745	−	0.897141i	$-0.354360\pi$
$739$	24.0173	0.883490	0.441745	+	0.897141i	$0.354360\pi$
$740$	−4.58936	−0.168708
$741$	0	0
$742$	−12.0832	−0.443587
$743$	−5.73799	−0.210507	−0.105253	−	0.994445i	$-0.533565\pi$
$743$	−5.73799	−0.210507	−0.105253	+	0.994445i	$0.533565\pi$
$744$	3.17142	0.116270
$745$	41.5164	1.52104
$746$	12.3028	0.450437
$747$	12.0081	0.439352
$748$	−11.4199	−0.417552
$749$	−19.9061	−0.727351
$750$	15.5320	0.567148
$751$	37.6807	1.37499	0.687495	−	0.726189i	$-0.258710\pi$
$751$	37.6807	1.37499	0.687495	+	0.726189i	$0.258710\pi$
$752$	5.77571	0.210618
$753$	18.8888	0.688344
$754$	−2.36213	−0.0860238
$755$	−39.0960	−1.42285
$756$	5.65667	0.205731
$757$	−39.0772	−1.42028	−0.710142	−	0.704059i	$-0.751369\pi$
$757$	−39.0772	−1.42028	−0.710142	+	0.704059i	$0.751369\pi$
$758$	9.72581	0.353257
$759$	27.4669	0.996984
$760$	0	0
$761$	13.4457	0.487407	0.243703	−	0.969850i	$-0.421638\pi$
$761$	13.4457	0.487407	0.243703	+	0.969850i	$0.421638\pi$
$762$	16.0902	0.582887
$763$	5.76507	0.208709
$764$	3.40620	0.123232
$765$	−10.3942	−0.375804
$766$	15.4098	0.556777
$767$	−6.60638	−0.238543
$768$	−1.40760	−0.0507925
$769$	−1.43300	−0.0516752	−0.0258376	−	0.999666i	$-0.508225\pi$
$769$	−1.43300	−0.0516752	−0.0258376	+	0.999666i	$0.508225\pi$
$770$	5.73752	0.206766
$771$	−9.24694	−0.333020
$772$	3.90426	0.140517
$773$	−33.2573	−1.19618	−0.598090	−	0.801429i	$-0.704074\pi$
$773$	−33.2573	−1.19618	−0.598090	+	0.801429i	$0.704074\pi$
$774$	1.13194	0.0406867
$775$	0.285236	0.0102460
$776$	−0.637330	−0.0228788
$777$	−2.85311	−0.102355
$778$	−2.58927	−0.0928297
$779$	0	0
$780$	4.18847	0.149971
$781$	29.8240	1.06719
$782$	−34.7033	−1.24099
$783$	10.1673	0.363349
$784$	1.00000	0.0357143
$785$	−17.8287	−0.636334
$786$	27.7789	0.990839
$787$	−12.5378	−0.446925	−0.223463	−	0.974713i	$-0.571736\pi$
$787$	−12.5378	−0.446925	−0.223463	+	0.974713i	$0.571736\pi$
$788$	−2.65223	−0.0944817
$789$	−40.2700	−1.43365
$790$	−6.23246	−0.221741
$791$	−10.7913	−0.383695
$792$	2.58128	0.0917217
$793$	−12.4149	−0.440867
$794$	−11.2929	−0.400769
$795$	38.5102	1.36582
$796$	12.4354	0.440760
$797$	2.86425	0.101457	0.0507285	−	0.998712i	$-0.483846\pi$
$797$	2.86425	0.101457	0.0507285	+	0.998712i	$0.483846\pi$
$798$	0	0
$799$	−26.0290	−0.920839
$800$	−0.126599	−0.00447597
$801$	13.6843	0.483509
$802$	−19.0880	−0.674021
$803$	−25.8812	−0.913327
$804$	−1.55976	−0.0550084
$805$	17.4355	0.614519
$806$	−2.96096	−0.104295
$807$	−27.7543	−0.976996
$808$	3.11677	0.109647
$809$	12.8448	0.451599	0.225799	−	0.974174i	$-0.427501\pi$
$809$	12.8448	0.451599	0.225799	+	0.974174i	$0.427501\pi$
$810$	−11.1091	−0.390333
$811$	−7.85076	−0.275678	−0.137839	−	0.990455i	$-0.544016\pi$
$811$	−7.85076	−0.275678	−0.137839	+	0.990455i	$0.544016\pi$
$812$	1.79740	0.0630763
$813$	−9.76539	−0.342487
$814$	−5.13627	−0.180026
$815$	−2.08969	−0.0731988
$816$	6.34355	0.222069
$817$	0	0
$818$	−30.7422	−1.07488
$819$	−1.33871	−0.0467782
$820$	18.6364	0.650810
$821$	−16.9053	−0.590000	−0.295000	−	0.955497i	$-0.595320\pi$
$821$	−16.9053	−0.590000	−0.295000	+	0.955497i	$0.595320\pi$
$822$	−24.5219	−0.855300
$823$	0.523333	0.0182422	0.00912112	−	0.999958i	$-0.497097\pi$
$823$	0.523333	0.0182422	0.00912112	+	0.999958i	$0.497097\pi$
$824$	−1.21127	−0.0421966
$825$	−0.451567	−0.0157215
$826$	5.02694	0.174910
$827$	−31.6821	−1.10170	−0.550848	−	0.834606i	$-0.685695\pi$
$827$	−31.6821	−1.10170	−0.550848	+	0.834606i	$0.685695\pi$
$828$	7.84411	0.272602
$829$	7.83907	0.272262	0.136131	−	0.990691i	$-0.456533\pi$
$829$	7.83907	0.272262	0.136131	+	0.990691i	$0.456533\pi$
$830$	−26.6909	−0.926454
$831$	31.2274	1.08327
$832$	1.31420	0.0455615
$833$	−4.50663	−0.156146
$834$	8.98685	0.311189
$835$	25.9734	0.898846
$836$	0	0
$837$	12.7448	0.440525
$838$	24.3506	0.841178
$839$	13.7773	0.475646	0.237823	−	0.971308i	$-0.423566\pi$
$839$	13.7773	0.475646	0.237823	+	0.971308i	$0.423566\pi$
$840$	−3.18710	−0.109965
$841$	−25.7694	−0.888599
$842$	−5.28504	−0.182134
$843$	−46.3094	−1.59498
$844$	14.5888	0.502166
$845$	25.5241	0.878055
$846$	5.88343	0.202276
$847$	−4.57875	−0.157328
$848$	12.0832	0.414938
$849$	3.49289	0.119876
$850$	0.570537	0.0195693
$851$	−15.6083	−0.535047
$852$	−16.5667	−0.567566
$853$	−46.1988	−1.58182	−0.790909	−	0.611934i	$-0.790392\pi$
$853$	−46.1988	−1.58182	−0.790909	+	0.611934i	$0.790392\pi$
$854$	9.44679	0.323262
$855$	0	0
$856$	19.9061	0.680375
$857$	24.2885	0.829681	0.414840	−	0.909894i	$-0.363837\pi$
$857$	24.2885	0.829681	0.414840	+	0.909894i	$0.363837\pi$
$858$	4.68760	0.160032
$859$	−52.9802	−1.80766	−0.903830	−	0.427892i	$-0.859257\pi$
$859$	−52.9802	−1.80766	−0.903830	+	0.427892i	$0.859257\pi$
$860$	−2.51601	−0.0857953
$861$	11.5858	0.394844
$862$	0.618814	0.0210769
$863$	−32.1746	−1.09524	−0.547618	−	0.836728i	$-0.684465\pi$
$863$	−32.1746	−1.09524	−0.547618	+	0.836728i	$0.684465\pi$
$864$	−5.65667	−0.192444
$865$	−2.30815	−0.0784793
$866$	36.8806	1.25326
$867$	−4.65879	−0.158221
$868$	2.25306	0.0764738
$869$	−6.97517	−0.236616
$870$	−5.72848	−0.194214
$871$	1.45625	0.0493432
$872$	−5.76507	−0.195230
$873$	−0.649217	−0.0219727
$874$	0	0
$875$	11.0344	0.373029
$876$	14.3766	0.485739
$877$	45.6282	1.54075	0.770377	−	0.637588i	$-0.220068\pi$
$877$	45.6282	1.54075	0.770377	+	0.637588i	$0.220068\pi$
$878$	39.8288	1.34416
$879$	−0.202014	−0.00681375
$880$	−5.73752	−0.193412
$881$	29.3106	0.987500	0.493750	−	0.869604i	$-0.335626\pi$
$881$	29.3106	0.987500	0.493750	+	0.869604i	$0.335626\pi$
$882$	1.01865	0.0342997
$883$	−39.9499	−1.34442	−0.672211	−	0.740360i	$-0.734655\pi$
$883$	−39.9499	−1.34442	−0.672211	+	0.740360i	$0.734655\pi$
$884$	−5.92260	−0.199198
$885$	−16.0213	−0.538551
$886$	−37.2011	−1.24980
$887$	−36.1050	−1.21229	−0.606144	−	0.795355i	$-0.707284\pi$
$887$	−36.1050	−1.21229	−0.606144	+	0.795355i	$0.707284\pi$
$888$	2.85311	0.0957441
$889$	11.4309	0.383381
$890$	−30.4166	−1.01957
$891$	−12.4329	−0.416518
$892$	−16.8285	−0.563459
$893$	0	0
$894$	−25.8099	−0.863211
$895$	−13.9675	−0.466881
$896$	−1.00000	−0.0334077
$897$	14.2449	0.475623
$898$	−27.2063	−0.907886
$899$	4.04964	0.135063
$900$	−0.128961	−0.00429869
$901$	−54.4544	−1.81414
$902$	20.8572	0.694469
$903$	−1.56415	−0.0520517
$904$	10.7913	0.358914
$905$	−40.7405	−1.35426
$906$	24.3052	0.807485
$907$	10.7645	0.357428	0.178714	−	0.983901i	$-0.442806\pi$
$907$	10.7645	0.357428	0.178714	+	0.983901i	$0.442806\pi$
$908$	−4.45322	−0.147785
$909$	3.17489	0.105305
$910$	2.97560	0.0986402
$911$	52.6695	1.74502	0.872509	−	0.488599i	$-0.162492\pi$
$911$	52.6695	1.74502	0.872509	+	0.488599i	$0.162492\pi$
$912$	0	0
$913$	−29.8715	−0.988604
$914$	−5.70914	−0.188842
$915$	−30.1078	−0.995334
$916$	20.1895	0.667081
$917$	19.7348	0.651702
$918$	25.4925	0.841379
$919$	47.3855	1.56310	0.781552	−	0.623840i	$-0.214428\pi$
$919$	47.3855	1.56310	0.781552	+	0.623840i	$0.214428\pi$
$920$	−17.4355	−0.574830
$921$	−32.8963	−1.08397
$922$	−18.4972	−0.609172
$923$	15.4673	0.509114
$924$	−3.56689	−0.117342
$925$	0.256608	0.00843721
$926$	−29.8307	−0.980298
$927$	−1.23386	−0.0405253
$928$	−1.79740	−0.0590025
$929$	46.9608	1.54073	0.770366	−	0.637602i	$-0.220073\pi$
$929$	46.9608	1.54073	0.770366	+	0.637602i	$0.220073\pi$
$930$	−7.18072	−0.235465
$931$	0	0
$932$	20.3141	0.665410
$933$	23.6588	0.774554
$934$	−34.9102	−1.14230
$935$	25.8569	0.845611
$936$	1.33871	0.0437570
$937$	4.08922	0.133589	0.0667945	−	0.997767i	$-0.478723\pi$
$937$	4.08922	0.133589	0.0667945	+	0.997767i	$0.478723\pi$
$938$	−1.10809	−0.0361806
$939$	−19.9386	−0.650673
$940$	−13.0774	−0.426536
$941$	5.65921	0.184485	0.0922425	−	0.995737i	$-0.470597\pi$
$941$	5.65921	0.184485	0.0922425	+	0.995737i	$0.470597\pi$
$942$	11.0837	0.361127
$943$	63.3819	2.06400
$944$	−5.02694	−0.163613
$945$	−12.8078	−0.416639
$946$	−2.81584	−0.0915507
$947$	−20.0115	−0.650286	−0.325143	−	0.945665i	$-0.605413\pi$
$947$	−20.0115	−0.650286	−0.325143	+	0.945665i	$0.605413\pi$
$948$	3.87459	0.125841
$949$	−13.4225	−0.435714
$950$	0	0
$951$	22.1210	0.717323
$952$	4.50663	0.146061
$953$	−24.3912	−0.790107	−0.395054	−	0.918658i	$-0.629274\pi$
$953$	−24.3912	−0.790107	−0.395054	+	0.918658i	$0.629274\pi$
$954$	12.3085	0.398503
$955$	−7.71233	−0.249565
$956$	25.8369	0.835625
$957$	−6.41113	−0.207242
$958$	−1.48027	−0.0478255
$959$	−17.4210	−0.562554
$960$	3.18710	0.102863
$961$	−25.9237	−0.836249
$962$	−2.66378	−0.0858836
$963$	20.2773	0.653427
$964$	16.3177	0.525558
$965$	−8.84002	−0.284570
$966$	−10.8393	−0.348747
$967$	34.3208	1.10368	0.551842	−	0.833949i	$-0.313925\pi$
$967$	34.3208	1.10368	0.551842	+	0.833949i	$0.313925\pi$
$968$	4.57875	0.147167
$969$	0	0
$970$	1.44304	0.0463333
$971$	8.68641	0.278760	0.139380	−	0.990239i	$-0.455489\pi$
$971$	8.68641	0.278760	0.139380	+	0.990239i	$0.455489\pi$
$972$	−10.0637	−0.322794
$973$	6.38450	0.204678
$974$	−29.1423	−0.933781
$975$	−0.234192	−0.00750016
$976$	−9.44679	−0.302384
$977$	33.2315	1.06317	0.531584	−	0.847005i	$-0.321597\pi$
$977$	33.2315	1.06317	0.531584	+	0.847005i	$0.321597\pi$
$978$	1.29912	0.0415412
$979$	−34.0413	−1.08796
$980$	−2.26420	−0.0723272
$981$	−5.87259	−0.187497
$982$	7.07800	0.225868
$983$	−13.4827	−0.430032	−0.215016	−	0.976611i	$-0.568980\pi$
$983$	−13.4827	−0.430032	−0.215016	+	0.976611i	$0.568980\pi$
$984$	−11.5858	−0.369343
$985$	6.00518	0.191341
$986$	8.10021	0.257963
$987$	−8.12991	−0.258778
$988$	0	0
$989$	−8.55690	−0.272094
$990$	−5.84453	−0.185751
$991$	1.59631	0.0507084	0.0253542	−	0.999679i	$-0.491929\pi$
$991$	1.59631	0.0507084	0.0253542	+	0.999679i	$0.491929\pi$
$992$	−2.25306	−0.0715347
$993$	11.0467	0.350557
$994$	−11.7694	−0.373304
$995$	−28.1562	−0.892610
$996$	16.5931	0.525774
$997$	−49.0430	−1.55321	−0.776603	−	0.629990i	$-0.783059\pi$
$997$	−49.0430	−1.55321	−0.776603	+	0.629990i	$0.783059\pi$
$998$	−11.0798	−0.350725
$999$	11.4656	0.362757

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bg.1.2	✓	8
19.18	odd	2		5054.2.a.bh.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bg.1.2	✓	8	1.1	even	1	trivial
5054.2.a.bh.1.7	yes	8	19.18	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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.40760 q^{3} +1.00000 q^{4} -2.26420 q^{5} +1.40760 q^{6} +1.00000 q^{7} -1.00000 q^{8} -1.01865 q^{9} +2.26420 q^{10} +2.53402 q^{11} -1.40760 q^{12} +1.31420 q^{13} -1.00000 q^{14} +3.18710 q^{15} +1.00000 q^{16} -4.50663 q^{17} +1.01865 q^{18} -2.26420 q^{20} -1.40760 q^{21} -2.53402 q^{22} -7.70050 q^{23} +1.40760 q^{24} +0.126599 q^{25} -1.31420 q^{26} +5.65667 q^{27} +1.00000 q^{28} +1.79740 q^{29} -3.18710 q^{30} +2.25306 q^{31} -1.00000 q^{32} -3.56689 q^{33} +4.50663 q^{34} -2.26420 q^{35} -1.01865 q^{36} +2.02693 q^{37} -1.84987 q^{39} +2.26420 q^{40} -8.23088 q^{41} +1.40760 q^{42} +1.11121 q^{43} +2.53402 q^{44} +2.30643 q^{45} +7.70050 q^{46} +5.77571 q^{47} -1.40760 q^{48} +1.00000 q^{49} -0.126599 q^{50} +6.34355 q^{51} +1.31420 q^{52} +12.0832 q^{53} -5.65667 q^{54} -5.73752 q^{55} -1.00000 q^{56} -1.79740 q^{58} -5.02694 q^{59} +3.18710 q^{60} -9.44679 q^{61} -2.25306 q^{62} -1.01865 q^{63} +1.00000 q^{64} -2.97560 q^{65} +3.56689 q^{66} +1.10809 q^{67} -4.50663 q^{68} +10.8393 q^{69} +2.26420 q^{70} +11.7694 q^{71} +1.01865 q^{72} -10.2135 q^{73} -2.02693 q^{74} -0.178202 q^{75} +2.53402 q^{77} +1.84987 q^{78} -2.75261 q^{79} -2.26420 q^{80} -4.90640 q^{81} +8.23088 q^{82} -11.7882 q^{83} -1.40760 q^{84} +10.2039 q^{85} -1.11121 q^{86} -2.53002 q^{87} -2.53402 q^{88} -13.4337 q^{89} -2.30643 q^{90} +1.31420 q^{91} -7.70050 q^{92} -3.17142 q^{93} -5.77571 q^{94} +1.40760 q^{96} +0.637330 q^{97} -1.00000 q^{98} -2.58128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 2 q^{10} - 12 q^{11} + 4 q^{12} + 10 q^{13} - 8 q^{14} + 24 q^{15} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 2 q^{20} + 4 q^{21} + 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} + 22 q^{27} + 8 q^{28} + 8 q^{29} - 24 q^{30} + 18 q^{31} - 8 q^{32} - 16 q^{33} + 6 q^{34} - 2 q^{35} + 8 q^{36} + 36 q^{37} + 2 q^{40} + 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 20 q^{46} - 10 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} + 12 q^{51} + 10 q^{52} + 32 q^{53} - 22 q^{54} - 22 q^{55} - 8 q^{56} - 8 q^{58} - 2 q^{59} + 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} + 44 q^{67} - 6 q^{68} + 2 q^{70} + 8 q^{71} - 8 q^{72} + 30 q^{73} - 36 q^{74} - 16 q^{75} - 12 q^{77} + 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} + 4 q^{84} - 16 q^{85} + 16 q^{86} + 24 q^{87} + 12 q^{88} - 22 q^{89} - 8 q^{90} + 10 q^{91} - 20 q^{92} - 16 q^{93} + 10 q^{94} - 4 q^{96} - 6 q^{97} - 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 2 * q^10 - 12 * q^11 + 4 * q^12 + 10 * q^13 - 8 * q^14 + 24 * q^15 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 2 * q^20 + 4 * q^21 + 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 + 22 * q^27 + 8 * q^28 + 8 * q^29 - 24 * q^30 + 18 * q^31 - 8 * q^32 - 16 * q^33 + 6 * q^34 - 2 * q^35 + 8 * q^36 + 36 * q^37 + 2 * q^40 + 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 + 20 * q^46 - 10 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 + 12 * q^51 + 10 * q^52 + 32 * q^53 - 22 * q^54 - 22 * q^55 - 8 * q^56 - 8 * q^58 - 2 * q^59 + 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 + 44 * q^67 - 6 * q^68 + 2 * q^70 + 8 * q^71 - 8 * q^72 + 30 * q^73 - 36 * q^74 - 16 * q^75 - 12 * q^77 + 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 + 4 * q^84 - 16 * q^85 + 16 * q^86 + 24 * q^87 + 12 * q^88 - 22 * q^89 - 8 * q^90 + 10 * q^91 - 20 * q^92 - 16 * q^93 + 10 * q^94 - 4 * q^96 - 6 * q^97 - 8 * q^98 - 52 * q^99

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