LMFDB - Embedded newform 5054.2.a.bi.1.3

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			1.3
Root			$-2.15124$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.578669 q^{3} +1.00000 q^{4} +1.36891 q^{5} -0.578669 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.66514 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.578669 q^{3} +1.00000 q^{4} +1.36891 q^{5} -0.578669 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.66514 q^{9} +1.36891 q^{10} +1.62784 q^{11} -0.578669 q^{12} -5.18824 q^{13} -1.00000 q^{14} -0.792143 q^{15} +1.00000 q^{16} +0.257918 q^{17} -2.66514 q^{18} +1.36891 q^{20} +0.578669 q^{21} +1.62784 q^{22} -0.275797 q^{23} -0.578669 q^{24} -3.12610 q^{25} -5.18824 q^{26} +3.27824 q^{27} -1.00000 q^{28} +3.15414 q^{29} -0.792143 q^{30} +7.06316 q^{31} +1.00000 q^{32} -0.941981 q^{33} +0.257918 q^{34} -1.36891 q^{35} -2.66514 q^{36} +9.35193 q^{37} +3.00228 q^{39} +1.36891 q^{40} +7.91308 q^{41} +0.578669 q^{42} -2.96944 q^{43} +1.62784 q^{44} -3.64833 q^{45} -0.275797 q^{46} -2.35659 q^{47} -0.578669 q^{48} +1.00000 q^{49} -3.12610 q^{50} -0.149249 q^{51} -5.18824 q^{52} +7.47738 q^{53} +3.27824 q^{54} +2.22836 q^{55} -1.00000 q^{56} +3.15414 q^{58} +2.57219 q^{59} -0.792143 q^{60} +12.3032 q^{61} +7.06316 q^{62} +2.66514 q^{63} +1.00000 q^{64} -7.10221 q^{65} -0.941981 q^{66} -3.59521 q^{67} +0.257918 q^{68} +0.159595 q^{69} -1.36891 q^{70} -3.93297 q^{71} -2.66514 q^{72} +1.56026 q^{73} +9.35193 q^{74} +1.80898 q^{75} -1.62784 q^{77} +3.00228 q^{78} +15.7365 q^{79} +1.36891 q^{80} +6.09841 q^{81} +7.91308 q^{82} +10.8657 q^{83} +0.578669 q^{84} +0.353065 q^{85} -2.96944 q^{86} -1.82520 q^{87} +1.62784 q^{88} +0.947298 q^{89} -3.64833 q^{90} +5.18824 q^{91} -0.275797 q^{92} -4.08723 q^{93} -2.35659 q^{94} -0.578669 q^{96} +4.32044 q^{97} +1.00000 q^{98} -4.33843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 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 + 6 * q^5 + 4 * q^6 - 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9} + 6 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 8 q^{14} + 8 q^{15} + 8 q^{16} + 2 q^{17} + 8 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} + 22 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} + 22 q^{31} + 8 q^{32} - 8 q^{33} + 2 q^{34} - 6 q^{35} + 8 q^{36} - 12 q^{37} + 8 q^{39} + 6 q^{40} + 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} + 20 q^{46} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} - 4 q^{51} + 6 q^{52} + 16 q^{53} + 22 q^{54} + 18 q^{55} - 8 q^{56} + 16 q^{58} + 14 q^{59} + 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} + 32 q^{65} - 8 q^{66} - 20 q^{67} + 2 q^{68} + 40 q^{69} - 6 q^{70} + 8 q^{72} - 18 q^{73} - 12 q^{74} + 16 q^{75} - 4 q^{77} + 8 q^{78} + 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} - 4 q^{84} - 16 q^{85} + 28 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{90} - 6 q^{91} + 20 q^{92} + 16 q^{93} + 6 q^{94} + 4 q^{96} + 26 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 + 8 * q^8 + 8 * q^9 + 6 * q^10 + 4 * q^11 + 4 * q^12 + 6 * q^13 - 8 * q^14 + 8 * q^15 + 8 * q^16 + 2 * q^17 + 8 * q^18 + 6 * q^20 - 4 * q^21 + 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 + 22 * q^27 - 8 * q^28 + 16 * q^29 + 8 * q^30 + 22 * q^31 + 8 * q^32 - 8 * q^33 + 2 * q^34 - 6 * q^35 + 8 * q^36 - 12 * q^37 + 8 * q^39 + 6 * q^40 + 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 + 20 * q^46 + 6 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 - 4 * q^51 + 6 * q^52 + 16 * q^53 + 22 * q^54 + 18 * q^55 - 8 * q^56 + 16 * q^58 + 14 * q^59 + 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 + 32 * q^65 - 8 * q^66 - 20 * q^67 + 2 * q^68 + 40 * q^69 - 6 * q^70 + 8 * q^72 - 18 * q^73 - 12 * q^74 + 16 * q^75 - 4 * q^77 + 8 * q^78 + 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 - 4 * q^84 - 16 * q^85 + 28 * q^87 + 4 * q^88 + 18 * q^89 - 4 * q^90 - 6 * q^91 + 20 * q^92 + 16 * q^93 + 6 * q^94 + 4 * q^96 + 26 * q^97 + 8 * q^98 + 4 * 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
$3$	−0.578669	−0.334095	−0.167047	−	0.985949i	$-0.553423\pi$
$3$	−0.578669	−0.334095	−0.167047	+	0.985949i	$0.553423\pi$
$4$	1.00000	0.500000
$5$	1.36891	0.612193	0.306097	−	0.952000i	$-0.400977\pi$
$5$	1.36891	0.612193	0.306097	+	0.952000i	$0.400977\pi$
$6$	−0.578669	−0.236241
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.66514	−0.888381
$10$	1.36891	0.432886
$11$	1.62784	0.490812	0.245406	−	0.969420i	$-0.421079\pi$
$11$	1.62784	0.490812	0.245406	+	0.969420i	$0.421079\pi$
$12$	−0.578669	−0.167047
$13$	−5.18824	−1.43896	−0.719480	−	0.694513i	$-0.755620\pi$
$13$	−5.18824	−1.43896	−0.719480	+	0.694513i	$0.755620\pi$
$14$	−1.00000	−0.267261
$15$	−0.792143	−0.204531
$16$	1.00000	0.250000
$17$	0.257918	0.0625543	0.0312771	−	0.999511i	$-0.490043\pi$
$17$	0.257918	0.0625543	0.0312771	+	0.999511i	$0.490043\pi$
$18$	−2.66514	−0.628180
$19$	0	0
$19$	0	0
$20$	1.36891	0.306097
$21$	0.578669	0.126276
$22$	1.62784	0.347057
$23$	−0.275797	−0.0575076	−0.0287538	−	0.999587i	$-0.509154\pi$
$23$	−0.275797	−0.0575076	−0.0287538	+	0.999587i	$0.509154\pi$
$24$	−0.578669	−0.118120
$25$	−3.12610	−0.625220
$26$	−5.18824	−1.01750
$27$	3.27824	0.630898
$28$	−1.00000	−0.188982
$29$	3.15414	0.585709	0.292854	−	0.956157i	$-0.405395\pi$
$29$	3.15414	0.585709	0.292854	+	0.956157i	$0.405395\pi$
$30$	−0.792143	−0.144625
$31$	7.06316	1.26858	0.634291	−	0.773095i	$-0.281292\pi$
$31$	7.06316	1.26858	0.634291	+	0.773095i	$0.281292\pi$
$32$	1.00000	0.176777
$33$	−0.941981	−0.163978
$34$	0.257918	0.0442326
$35$	−1.36891	−0.231387
$36$	−2.66514	−0.444190
$37$	9.35193	1.53745	0.768724	−	0.639581i	$-0.220892\pi$
$37$	9.35193	1.53745	0.768724	+	0.639581i	$0.220892\pi$
$38$	0	0
$39$	3.00228	0.480749
$40$	1.36891	0.216443
$41$	7.91308	1.23582	0.617908	−	0.786251i	$-0.287980\pi$
$41$	7.91308	1.23582	0.617908	+	0.786251i	$0.287980\pi$
$42$	0.578669	0.0892906
$43$	−2.96944	−0.452835	−0.226417	−	0.974030i	$-0.572701\pi$
$43$	−2.96944	−0.452835	−0.226417	+	0.974030i	$0.572701\pi$
$44$	1.62784	0.245406
$45$	−3.64833	−0.543860
$46$	−0.275797	−0.0406640
$47$	−2.35659	−0.343744	−0.171872	−	0.985119i	$-0.554982\pi$
$47$	−2.35659	−0.343744	−0.171872	+	0.985119i	$0.554982\pi$
$48$	−0.578669	−0.0835237
$49$	1.00000	0.142857
$50$	−3.12610	−0.442097
$51$	−0.149249	−0.0208991
$52$	−5.18824	−0.719480
$53$	7.47738	1.02710	0.513549	−	0.858060i	$-0.328331\pi$
$53$	7.47738	1.02710	0.513549	+	0.858060i	$0.328331\pi$
$54$	3.27824	0.446112
$55$	2.22836	0.300472
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.15414	0.414159
$59$	2.57219	0.334871	0.167435	−	0.985883i	$-0.446451\pi$
$59$	2.57219	0.334871	0.167435	+	0.985883i	$0.446451\pi$
$60$	−0.792143	−0.102265
$61$	12.3032	1.57526	0.787630	−	0.616149i	$-0.211308\pi$
$61$	12.3032	1.57526	0.787630	+	0.616149i	$0.211308\pi$
$62$	7.06316	0.897022
$63$	2.66514	0.335776
$64$	1.00000	0.125000
$65$	−7.10221	−0.880921
$66$	−0.941981	−0.115950
$67$	−3.59521	−0.439225	−0.219612	−	0.975587i	$-0.570479\pi$
$67$	−3.59521	−0.439225	−0.219612	+	0.975587i	$0.570479\pi$
$68$	0.257918	0.0312771
$69$	0.159595	0.0192130
$70$	−1.36891	−0.163615
$71$	−3.93297	−0.466758	−0.233379	−	0.972386i	$-0.574978\pi$
$71$	−3.93297	−0.466758	−0.233379	+	0.972386i	$0.574978\pi$
$72$	−2.66514	−0.314090
$73$	1.56026	0.182614	0.0913072	−	0.995823i	$-0.470895\pi$
$73$	1.56026	0.182614	0.0913072	+	0.995823i	$0.470895\pi$
$74$	9.35193	1.08714
$75$	1.80898	0.208883
$76$	0	0
$77$	−1.62784	−0.185510
$78$	3.00228	0.339941
$79$	15.7365	1.77050	0.885248	−	0.465120i	$-0.153989\pi$
$79$	15.7365	1.77050	0.885248	+	0.465120i	$0.153989\pi$
$80$	1.36891	0.153048
$81$	6.09841	0.677601
$82$	7.91308	0.873854
$83$	10.8657	1.19266	0.596331	−	0.802739i	$-0.296625\pi$
$83$	10.8657	1.19266	0.596331	+	0.802739i	$0.296625\pi$
$84$	0.578669	0.0631380
$85$	0.353065	0.0382953
$86$	−2.96944	−0.320202
$87$	−1.82520	−0.195682
$88$	1.62784	0.173528
$89$	0.947298	0.100413	0.0502067	−	0.998739i	$-0.484012\pi$
$89$	0.947298	0.100413	0.0502067	+	0.998739i	$0.484012\pi$
$90$	−3.64833	−0.384567
$91$	5.18824	0.543876
$92$	−0.275797	−0.0287538
$93$	−4.08723	−0.423826
$94$	−2.35659	−0.243064
$95$	0	0
$96$	−0.578669	−0.0590602
$97$	4.32044	0.438675	0.219337	−	0.975649i	$-0.429610\pi$
$97$	4.32044	0.438675	0.219337	+	0.975649i	$0.429610\pi$
$98$	1.00000	0.101015
$99$	−4.33843	−0.436028
$100$	−3.12610	−0.312610
$101$	12.0383	1.19786	0.598928	−	0.800803i	$-0.295593\pi$
$101$	12.0383	1.19786	0.598928	+	0.800803i	$0.295593\pi$
$102$	−0.149249	−0.0147779
$103$	5.10605	0.503114	0.251557	−	0.967842i	$-0.419057\pi$
$103$	5.10605	0.503114	0.251557	+	0.967842i	$0.419057\pi$
$104$	−5.18824	−0.508749
$105$	0.792143	0.0773053
$106$	7.47738	0.726268
$107$	−4.21439	−0.407420	−0.203710	−	0.979031i	$-0.565300\pi$
$107$	−4.21439	−0.407420	−0.203710	+	0.979031i	$0.565300\pi$
$108$	3.27824	0.315449
$109$	−12.3453	−1.18246	−0.591232	−	0.806502i	$-0.701358\pi$
$109$	−12.3453	−1.18246	−0.591232	+	0.806502i	$0.701358\pi$
$110$	2.22836	0.212466
$111$	−5.41167	−0.513653
$112$	−1.00000	−0.0944911
$113$	15.3482	1.44384	0.721918	−	0.691979i	$-0.243261\pi$
$113$	15.3482	1.44384	0.721918	+	0.691979i	$0.243261\pi$
$114$	0	0
$115$	−0.377540	−0.0352058
$116$	3.15414	0.292854
$117$	13.8274	1.27834
$118$	2.57219	0.236789
$119$	−0.257918	−0.0236433
$120$	−0.792143	−0.0723125
$121$	−8.35013	−0.759103
$122$	12.3032	1.11388
$123$	−4.57906	−0.412880
$124$	7.06316	0.634291
$125$	−11.1239	−0.994948
$126$	2.66514	0.237430
$127$	−12.5502	−1.11365	−0.556824	−	0.830630i	$-0.687980\pi$
$127$	−12.5502	−1.11365	−0.556824	+	0.830630i	$0.687980\pi$
$128$	1.00000	0.0883883
$129$	1.71832	0.151290
$130$	−7.10221	−0.622905
$131$	8.20043	0.716475	0.358238	−	0.933630i	$-0.383378\pi$
$131$	8.20043	0.716475	0.358238	+	0.933630i	$0.383378\pi$
$132$	−0.941981	−0.0819890
$133$	0	0
$134$	−3.59521	−0.310579
$135$	4.48760	0.386231
$136$	0.257918	0.0221163
$137$	6.57313	0.561580	0.280790	−	0.959769i	$-0.409403\pi$
$137$	6.57313	0.561580	0.280790	+	0.959769i	$0.409403\pi$
$138$	0.159595	0.0135856
$139$	−2.58506	−0.219262	−0.109631	−	0.993972i	$-0.534967\pi$
$139$	−2.58506	−0.219262	−0.109631	+	0.993972i	$0.534967\pi$
$140$	−1.36891	−0.115694
$141$	1.36369	0.114843
$142$	−3.93297	−0.330048
$143$	−8.44563	−0.706259
$144$	−2.66514	−0.222095
$145$	4.31772	0.358567
$146$	1.56026	0.129128
$147$	−0.578669	−0.0477278
$148$	9.35193	0.768724
$149$	−16.4215	−1.34530	−0.672650	−	0.739960i	$-0.734844\pi$
$149$	−16.4215	−1.34530	−0.672650	+	0.739960i	$0.734844\pi$
$150$	1.80898	0.147702
$151$	1.43635	0.116889	0.0584443	−	0.998291i	$-0.481386\pi$
$151$	1.43635	0.116889	0.0584443	+	0.998291i	$0.481386\pi$
$152$	0	0
$153$	−0.687388	−0.0555720
$154$	−1.62784	−0.131175
$155$	9.66880	0.776617
$156$	3.00228	0.240374
$157$	−14.7226	−1.17499	−0.587495	−	0.809228i	$-0.699886\pi$
$157$	−14.7226	−1.17499	−0.587495	+	0.809228i	$0.699886\pi$
$158$	15.7365	1.25193
$159$	−4.32693	−0.343148
$160$	1.36891	0.108221
$161$	0.275797	0.0217358
$162$	6.09841	0.479136
$163$	−23.3097	−1.82576	−0.912878	−	0.408234i	$-0.866145\pi$
$163$	−23.3097	−1.82576	−0.912878	+	0.408234i	$0.866145\pi$
$164$	7.91308	0.617908
$165$	−1.28948	−0.100386
$166$	10.8657	0.843340
$167$	−17.9819	−1.39148	−0.695740	−	0.718294i	$-0.744923\pi$
$167$	−17.9819	−1.39148	−0.695740	+	0.718294i	$0.744923\pi$
$168$	0.578669	0.0446453
$169$	13.9179	1.07060
$170$	0.353065	0.0270789
$171$	0	0
$172$	−2.96944	−0.226417
$173$	14.5104	1.10321	0.551603	−	0.834107i	$-0.314016\pi$
$173$	14.5104	1.10321	0.551603	+	0.834107i	$0.314016\pi$
$174$	−1.82520	−0.138368
$175$	3.12610	0.236311
$176$	1.62784	0.122703
$177$	−1.48845	−0.111879
$178$	0.947298	0.0710030
$179$	22.6246	1.69104	0.845522	−	0.533941i	$-0.179289\pi$
$179$	22.6246	1.69104	0.845522	+	0.533941i	$0.179289\pi$
$180$	−3.64833	−0.271930
$181$	−26.7353	−1.98722	−0.993608	−	0.112883i	$-0.963992\pi$
$181$	−26.7353	−1.98722	−0.993608	+	0.112883i	$0.963992\pi$
$182$	5.18824	0.384578
$183$	−7.11946	−0.526286
$184$	−0.275797	−0.0203320
$185$	12.8019	0.941215
$186$	−4.08723	−0.299691
$187$	0.419849	0.0307024
$188$	−2.35659	−0.171872
$189$	−3.27824	−0.238457
$190$	0	0
$191$	11.2051	0.810772	0.405386	−	0.914146i	$-0.367137\pi$
$191$	11.2051	0.810772	0.405386	+	0.914146i	$0.367137\pi$
$192$	−0.578669	−0.0417619
$193$	1.14499	0.0824185	0.0412092	−	0.999151i	$-0.486879\pi$
$193$	1.14499	0.0824185	0.0412092	+	0.999151i	$0.486879\pi$
$194$	4.32044	0.310190
$195$	4.10983	0.294311
$196$	1.00000	0.0714286
$197$	−4.23216	−0.301529	−0.150764	−	0.988570i	$-0.548173\pi$
$197$	−4.23216	−0.301529	−0.150764	+	0.988570i	$0.548173\pi$
$198$	−4.33843	−0.308319
$199$	−11.1242	−0.788572	−0.394286	−	0.918988i	$-0.629008\pi$
$199$	−11.1242	−0.788572	−0.394286	+	0.918988i	$0.629008\pi$
$200$	−3.12610	−0.221049
$201$	2.08044	0.146743
$202$	12.0383	0.847013
$203$	−3.15414	−0.221377
$204$	−0.149249	−0.0104495
$205$	10.8323	0.756558
$206$	5.10605	0.355755
$207$	0.735038	0.0510887
$208$	−5.18824	−0.359740
$209$	0	0
$210$	0.792143	0.0546631
$211$	26.2736	1.80875	0.904375	−	0.426738i	$-0.140337\pi$
$211$	26.2736	1.80875	0.904375	+	0.426738i	$0.140337\pi$
$212$	7.47738	0.513549
$213$	2.27589	0.155941
$214$	−4.21439	−0.288089
$215$	−4.06488	−0.277222
$216$	3.27824	0.223056
$217$	−7.06316	−0.479479
$218$	−12.3453	−0.836128
$219$	−0.902873	−0.0610105
$220$	2.22836	0.150236
$221$	−1.33814	−0.0900131
$222$	−5.41167	−0.363208
$223$	13.1498	0.880574	0.440287	−	0.897857i	$-0.354877\pi$
$223$	13.1498	0.880574	0.440287	+	0.897857i	$0.354877\pi$
$224$	−1.00000	−0.0668153
$225$	8.33150	0.555433
$226$	15.3482	1.02095
$227$	−4.75659	−0.315706	−0.157853	−	0.987463i	$-0.550457\pi$
$227$	−4.75659	−0.315706	−0.157853	+	0.987463i	$0.550457\pi$
$228$	0	0
$229$	−10.6582	−0.704313	−0.352156	−	0.935941i	$-0.614551\pi$
$229$	−10.6582	−0.704313	−0.352156	+	0.935941i	$0.614551\pi$
$230$	−0.377540	−0.0248942
$231$	0.941981	0.0619778
$232$	3.15414	0.207079
$233$	1.58547	0.103868	0.0519339	−	0.998651i	$-0.483461\pi$
$233$	1.58547	0.103868	0.0519339	+	0.998651i	$0.483461\pi$
$234$	13.8274	0.903926
$235$	−3.22595	−0.210438
$236$	2.57219	0.167435
$237$	−9.10623	−0.591513
$238$	−0.257918	−0.0167183
$239$	−13.6183	−0.880894	−0.440447	−	0.897779i	$-0.645180\pi$
$239$	−13.6183	−0.880894	−0.440447	+	0.897779i	$0.645180\pi$
$240$	−0.792143	−0.0511326
$241$	2.51719	0.162147	0.0810734	−	0.996708i	$-0.474165\pi$
$241$	2.51719	0.162147	0.0810734	+	0.996708i	$0.474165\pi$
$242$	−8.35013	−0.536767
$243$	−13.3637	−0.857281
$244$	12.3032	0.787630
$245$	1.36891	0.0874562
$246$	−4.57906	−0.291950
$247$	0	0
$248$	7.06316	0.448511
$249$	−6.28763	−0.398462
$250$	−11.1239	−0.703535
$251$	15.5613	0.982219	0.491109	−	0.871098i	$-0.336592\pi$
$251$	15.5613	0.982219	0.491109	+	0.871098i	$0.336592\pi$
$252$	2.66514	0.167888
$253$	−0.448953	−0.0282255
$254$	−12.5502	−0.787468
$255$	−0.204308	−0.0127943
$256$	1.00000	0.0625000
$257$	17.4424	1.08803	0.544013	−	0.839077i	$-0.316904\pi$
$257$	17.4424	1.08803	0.544013	+	0.839077i	$0.316904\pi$
$258$	1.71832	0.106978
$259$	−9.35193	−0.581101
$260$	−7.10221	−0.440461
$261$	−8.40622	−0.520332
$262$	8.20043	0.506624
$263$	−0.698187	−0.0430521	−0.0215260	−	0.999768i	$-0.506852\pi$
$263$	−0.698187	−0.0430521	−0.0215260	+	0.999768i	$0.506852\pi$
$264$	−0.941981	−0.0579749
$265$	10.2358	0.628782
$266$	0	0
$267$	−0.548172	−0.0335476
$268$	−3.59521	−0.219612
$269$	16.1173	0.982688	0.491344	−	0.870966i	$-0.336506\pi$
$269$	16.1173	0.982688	0.491344	+	0.870966i	$0.336506\pi$
$270$	4.48760	0.273107
$271$	20.6217	1.25268	0.626340	−	0.779550i	$-0.284552\pi$
$271$	20.6217	1.25268	0.626340	+	0.779550i	$0.284552\pi$
$272$	0.257918	0.0156386
$273$	−3.00228	−0.181706
$274$	6.57313	0.397097
$275$	−5.08879	−0.306866
$276$	0.159595	0.00960650
$277$	−26.5364	−1.59442	−0.797208	−	0.603705i	$-0.793690\pi$
$277$	−26.5364	−1.59442	−0.797208	+	0.603705i	$0.793690\pi$
$278$	−2.58506	−0.155042
$279$	−18.8243	−1.12698
$280$	−1.36891	−0.0818077
$281$	8.20137	0.489253	0.244626	−	0.969617i	$-0.421335\pi$
$281$	8.20137	0.489253	0.244626	+	0.969617i	$0.421335\pi$
$282$	1.36369	0.0812064
$283$	13.1749	0.783165	0.391582	−	0.920143i	$-0.371928\pi$
$283$	13.1749	0.783165	0.391582	+	0.920143i	$0.371928\pi$
$284$	−3.93297	−0.233379
$285$	0	0
$286$	−8.44563	−0.499401
$287$	−7.91308	−0.467094
$288$	−2.66514	−0.157045
$289$	−16.9335	−0.996087
$290$	4.31772	0.253545
$291$	−2.50011	−0.146559
$292$	1.56026	0.0913072
$293$	7.60243	0.444139	0.222069	−	0.975031i	$-0.428719\pi$
$293$	7.60243	0.444139	0.222069	+	0.975031i	$0.428719\pi$
$294$	−0.578669	−0.0337487
$295$	3.52109	0.205006
$296$	9.35193	0.543570
$297$	5.33646	0.309653
$298$	−16.4215	−0.951271
$299$	1.43090	0.0827511
$300$	1.80898	0.104441
$301$	2.96944	0.171155
$302$	1.43635	0.0826527
$303$	−6.96620	−0.400198
$304$	0	0
$305$	16.8419	0.964363
$306$	−0.687388	−0.0392954
$307$	−21.2034	−1.21014	−0.605072	−	0.796170i	$-0.706856\pi$
$307$	−21.2034	−1.21014	−0.605072	+	0.796170i	$0.706856\pi$
$308$	−1.62784	−0.0927548
$309$	−2.95471	−0.168088
$310$	9.66880	0.549151
$311$	31.5530	1.78921	0.894604	−	0.446859i	$-0.147458\pi$
$311$	31.5530	1.78921	0.894604	+	0.446859i	$0.147458\pi$
$312$	3.00228	0.169970
$313$	2.14974	0.121510	0.0607552	−	0.998153i	$-0.480649\pi$
$313$	2.14974	0.121510	0.0607552	+	0.998153i	$0.480649\pi$
$314$	−14.7226	−0.830843
$315$	3.64833	0.205560
$316$	15.7365	0.885248
$317$	−8.35113	−0.469046	−0.234523	−	0.972111i	$-0.575353\pi$
$317$	−8.35113	−0.469046	−0.234523	+	0.972111i	$0.575353\pi$
$318$	−4.32693	−0.242642
$319$	5.13443	0.287473
$320$	1.36891	0.0765241
$321$	2.43873	0.136117
$322$	0.275797	0.0153696
$323$	0	0
$324$	6.09841	0.338800
$325$	16.2190	0.899666
$326$	−23.3097	−1.29100
$327$	7.14383	0.395055
$328$	7.91308	0.436927
$329$	2.35659	0.129923
$330$	−1.28948	−0.0709837
$331$	−4.16252	−0.228793	−0.114396	−	0.993435i	$-0.536493\pi$
$331$	−4.16252	−0.228793	−0.114396	+	0.993435i	$0.536493\pi$
$332$	10.8657	0.596331
$333$	−24.9242	−1.36584
$334$	−17.9819	−0.983925
$335$	−4.92150	−0.268890
$336$	0.578669	0.0315690
$337$	−4.26524	−0.232342	−0.116171	−	0.993229i	$-0.537062\pi$
$337$	−4.26524	−0.232342	−0.116171	+	0.993229i	$0.537062\pi$
$338$	13.9179	0.757032
$339$	−8.88152	−0.482378
$340$	0.353065	0.0191477
$341$	11.4977	0.622636
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−2.96944	−0.160101
$345$	0.218471	0.0117621
$346$	14.5104	0.780085
$347$	20.2296	1.08598	0.542991	−	0.839739i	$-0.317292\pi$
$347$	20.2296	1.08598	0.542991	+	0.839739i	$0.317292\pi$
$348$	−1.82520	−0.0978411
$349$	7.80705	0.417902	0.208951	−	0.977926i	$-0.432995\pi$
$349$	7.80705	0.417902	0.208951	+	0.977926i	$0.432995\pi$
$350$	3.12610	0.167097
$351$	−17.0083	−0.907837
$352$	1.62784	0.0867642
$353$	−25.5973	−1.36241	−0.681204	−	0.732093i	$-0.738543\pi$
$353$	−25.5973	−1.36241	−0.681204	+	0.732093i	$0.738543\pi$
$354$	−1.48845	−0.0791101
$355$	−5.38386	−0.285746
$356$	0.947298	0.0502067
$357$	0.149249	0.00789910
$358$	22.6246	1.19575
$359$	−12.1661	−0.642103	−0.321052	−	0.947062i	$-0.604036\pi$
$359$	−12.1661	−0.642103	−0.321052	+	0.947062i	$0.604036\pi$
$360$	−3.64833	−0.192284
$361$	0	0
$362$	−26.7353	−1.40517
$363$	4.83197	0.253612
$364$	5.18824	0.271938
$365$	2.13585	0.111795
$366$	−7.11946	−0.372140
$367$	2.68751	0.140287	0.0701435	−	0.997537i	$-0.477654\pi$
$367$	2.68751	0.140287	0.0701435	+	0.997537i	$0.477654\pi$
$368$	−0.275797	−0.0143769
$369$	−21.0895	−1.09787
$370$	12.8019	0.665539
$371$	−7.47738	−0.388206
$372$	−4.08723	−0.211913
$373$	8.54484	0.442435	0.221218	−	0.975224i	$-0.428997\pi$
$373$	8.54484	0.442435	0.221218	+	0.975224i	$0.428997\pi$
$374$	0.419849	0.0217099
$375$	6.43703	0.332407
$376$	−2.35659	−0.121532
$377$	−16.3644	−0.842811
$378$	−3.27824	−0.168615
$379$	31.1455	1.59984	0.799919	−	0.600108i	$-0.204876\pi$
$379$	31.1455	1.59984	0.799919	+	0.600108i	$0.204876\pi$
$380$	0	0
$381$	7.26240	0.372064
$382$	11.2051	0.573303
$383$	25.6663	1.31149	0.655743	−	0.754984i	$-0.272356\pi$
$383$	25.6663	1.31149	0.655743	+	0.754984i	$0.272356\pi$
$384$	−0.578669	−0.0295301
$385$	−2.22836	−0.113568
$386$	1.14499	0.0582786
$387$	7.91397	0.402290
$388$	4.32044	0.219337
$389$	−13.6211	−0.690615	−0.345307	−	0.938490i	$-0.612225\pi$
$389$	−13.6211	−0.690615	−0.345307	+	0.938490i	$0.612225\pi$
$390$	4.10983	0.208109
$391$	−0.0711330	−0.00359735
$392$	1.00000	0.0505076
$393$	−4.74534	−0.239371
$394$	−4.23216	−0.213213
$395$	21.5418	1.08389
$396$	−4.33843	−0.218014
$397$	−13.8436	−0.694791	−0.347396	−	0.937719i	$-0.612934\pi$
$397$	−13.8436	−0.694791	−0.347396	+	0.937719i	$0.612934\pi$
$398$	−11.1242	−0.557605
$399$	0	0
$400$	−3.12610	−0.156305
$401$	−36.7080	−1.83311	−0.916556	−	0.399906i	$-0.869043\pi$
$401$	−36.7080	−1.83311	−0.916556	+	0.399906i	$0.869043\pi$
$402$	2.08044	0.103763
$403$	−36.6454	−1.82544
$404$	12.0383	0.598928
$405$	8.34814	0.414823
$406$	−3.15414	−0.156537
$407$	15.2235	0.754599
$408$	−0.149249	−0.00738894
$409$	−10.2376	−0.506219	−0.253110	−	0.967438i	$-0.581453\pi$
$409$	−10.2376	−0.506219	−0.253110	+	0.967438i	$0.581453\pi$
$410$	10.8323	0.534967
$411$	−3.80367	−0.187621
$412$	5.10605	0.251557
$413$	−2.57219	−0.126569
$414$	0.735038	0.0361251
$415$	14.8741	0.730140
$416$	−5.18824	−0.254375
$417$	1.49590	0.0732544
$418$	0	0
$419$	16.2221	0.792502	0.396251	−	0.918142i	$-0.370311\pi$
$419$	16.2221	0.792502	0.396251	+	0.918142i	$0.370311\pi$
$420$	0.792143	0.0386526
$421$	11.8248	0.576305	0.288152	−	0.957585i	$-0.406959\pi$
$421$	11.8248	0.576305	0.288152	+	0.957585i	$0.406959\pi$
$422$	26.2736	1.27898
$423$	6.28065	0.305376
$424$	7.47738	0.363134
$425$	−0.806277	−0.0391102
$426$	2.27589	0.110267
$427$	−12.3032	−0.595392
$428$	−4.21439	−0.203710
$429$	4.88723	0.235958
$430$	−4.06488	−0.196026
$431$	37.1721	1.79052	0.895259	−	0.445546i	$-0.146991\pi$
$431$	37.1721	1.79052	0.895259	+	0.445546i	$0.146991\pi$
$432$	3.27824	0.157725
$433$	−1.18587	−0.0569892	−0.0284946	−	0.999594i	$-0.509071\pi$
$433$	−1.18587	−0.0569892	−0.0284946	+	0.999594i	$0.509071\pi$
$434$	−7.06316	−0.339043
$435$	−2.49853	−0.119795
$436$	−12.3453	−0.591232
$437$	0	0
$438$	−0.902873	−0.0431410
$439$	28.6657	1.36814	0.684069	−	0.729417i	$-0.260209\pi$
$439$	28.6657	1.36814	0.684069	+	0.729417i	$0.260209\pi$
$440$	2.22836	0.106233
$441$	−2.66514	−0.126912
$442$	−1.33814	−0.0636489
$443$	25.2457	1.19946	0.599731	−	0.800202i	$-0.295274\pi$
$443$	25.2457	1.19946	0.599731	+	0.800202i	$0.295274\pi$
$444$	−5.41167	−0.256827
$445$	1.29676	0.0614724
$446$	13.1498	0.622660
$447$	9.50261	0.449458
$448$	−1.00000	−0.0472456
$449$	−10.2395	−0.483230	−0.241615	−	0.970372i	$-0.577677\pi$
$449$	−10.2395	−0.483230	−0.241615	+	0.970372i	$0.577677\pi$
$450$	8.33150	0.392750
$451$	12.8812	0.606554
$452$	15.3482	0.721918
$453$	−0.831172	−0.0390518
$454$	−4.75659	−0.223238
$455$	7.10221	0.332957
$456$	0	0
$457$	−14.5779	−0.681927	−0.340964	−	0.940076i	$-0.610753\pi$
$457$	−14.5779	−0.681927	−0.340964	+	0.940076i	$0.610753\pi$
$458$	−10.6582	−0.498024
$459$	0.845518	0.0394654
$460$	−0.377540	−0.0176029
$461$	34.8330	1.62234	0.811168	−	0.584814i	$-0.198832\pi$
$461$	34.8330	1.62234	0.811168	+	0.584814i	$0.198832\pi$
$462$	0.941981	0.0438249
$463$	−0.367718	−0.0170893	−0.00854465	−	0.999963i	$-0.502720\pi$
$463$	−0.367718	−0.0170893	−0.00854465	+	0.999963i	$0.502720\pi$
$464$	3.15414	0.146427
$465$	−5.59504	−0.259464
$466$	1.58547	0.0734456
$467$	−19.4832	−0.901574	−0.450787	−	0.892632i	$-0.648857\pi$
$467$	−19.4832	−0.901574	−0.450787	+	0.892632i	$0.648857\pi$
$468$	13.8274	0.639172
$469$	3.59521	0.166011
$470$	−3.22595	−0.148802
$471$	8.51950	0.392558
$472$	2.57219	0.118395
$473$	−4.83377	−0.222257
$474$	−9.10623	−0.418263
$475$	0	0
$476$	−0.257918	−0.0118217
$477$	−19.9283	−0.912454
$478$	−13.6183	−0.622886
$479$	−30.7828	−1.40650	−0.703250	−	0.710943i	$-0.748269\pi$
$479$	−30.7828	−1.40650	−0.703250	+	0.710943i	$0.748269\pi$
$480$	−0.792143	−0.0361562
$481$	−48.5201	−2.21233
$482$	2.51719	0.114655
$483$	−0.159595	−0.00726183
$484$	−8.35013	−0.379552
$485$	5.91428	0.268554
$486$	−13.3637	−0.606189
$487$	−21.5085	−0.974643	−0.487321	−	0.873223i	$-0.662026\pi$
$487$	−21.5085	−0.974643	−0.487321	+	0.873223i	$0.662026\pi$
$488$	12.3032	0.556938
$489$	13.4886	0.609975
$490$	1.36891	0.0618408
$491$	6.94462	0.313406	0.156703	−	0.987646i	$-0.449913\pi$
$491$	6.94462	0.313406	0.156703	+	0.987646i	$0.449913\pi$
$492$	−4.57906	−0.206440
$493$	0.813509	0.0366386
$494$	0	0
$495$	−5.93890	−0.266934
$496$	7.06316	0.317145
$497$	3.93297	0.176418
$498$	−6.28763	−0.281755
$499$	−16.0613	−0.719003	−0.359502	−	0.933145i	$-0.617053\pi$
$499$	−16.0613	−0.719003	−0.359502	+	0.933145i	$0.617053\pi$
$500$	−11.1239	−0.497474
$501$	10.4056	0.464886
$502$	15.5613	0.694534
$503$	24.9929	1.11438	0.557190	−	0.830385i	$-0.311879\pi$
$503$	24.9929	1.11438	0.557190	+	0.830385i	$0.311879\pi$
$504$	2.66514	0.118715
$505$	16.4793	0.733320
$506$	−0.448953	−0.0199584
$507$	−8.05384	−0.357683
$508$	−12.5502	−0.556824
$509$	−5.52919	−0.245077	−0.122538	−	0.992464i	$-0.539103\pi$
$509$	−5.52919	−0.245077	−0.122538	+	0.992464i	$0.539103\pi$
$510$	−0.204308	−0.00904691
$511$	−1.56026	−0.0690218
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.4424	0.769350
$515$	6.98970	0.308003
$516$	1.71832	0.0756449
$517$	−3.83616	−0.168714
$518$	−9.35193	−0.410900
$519$	−8.39673	−0.368576
$520$	−7.10221	−0.311453
$521$	−0.397939	−0.0174340	−0.00871701	−	0.999962i	$-0.502775\pi$
$521$	−0.397939	−0.0174340	−0.00871701	+	0.999962i	$0.502775\pi$
$522$	−8.40622	−0.367930
$523$	14.7312	0.644149	0.322074	−	0.946714i	$-0.395620\pi$
$523$	14.7312	0.644149	0.322074	+	0.946714i	$0.395620\pi$
$524$	8.20043	0.358238
$525$	−1.80898	−0.0789502
$526$	−0.698187	−0.0304424
$527$	1.82172	0.0793552
$528$	−0.941981	−0.0409945
$529$	−22.9239	−0.996693
$530$	10.2358	0.444616
$531$	−6.85526	−0.297493
$532$	0	0
$533$	−41.0550	−1.77829
$534$	−0.548172	−0.0237217
$535$	−5.76909	−0.249420
$536$	−3.59521	−0.155289
$537$	−13.0922	−0.564969
$538$	16.1173	0.694865
$539$	1.62784	0.0701161
$540$	4.48760	0.193116
$541$	−42.3044	−1.81881	−0.909403	−	0.415915i	$-0.863461\pi$
$541$	−42.3044	−1.81881	−0.909403	+	0.415915i	$0.863461\pi$
$542$	20.6217	0.885778
$543$	15.4709	0.663919
$544$	0.257918	0.0110581
$545$	−16.8995	−0.723896
$546$	−3.00228	−0.128486
$547$	−8.11438	−0.346946	−0.173473	−	0.984839i	$-0.555499\pi$
$547$	−8.11438	−0.346946	−0.173473	+	0.984839i	$0.555499\pi$
$548$	6.57313	0.280790
$549$	−32.7897	−1.39943
$550$	−5.08879	−0.216987
$551$	0	0
$552$	0.159595	0.00679282
$553$	−15.7365	−0.669184
$554$	−26.5364	−1.12742
$555$	−7.40807	−0.314455
$556$	−2.58506	−0.109631
$557$	−9.74617	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$557$	−9.74617	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$558$	−18.8243	−0.796897
$559$	15.4062	0.651611
$560$	−1.36891	−0.0578468
$561$	−0.242954	−0.0102575
$562$	8.20137	0.345954
$563$	−25.7838	−1.08666	−0.543328	−	0.839520i	$-0.682836\pi$
$563$	−25.7838	−1.08666	−0.543328	+	0.839520i	$0.682836\pi$
$564$	1.36369	0.0574216
$565$	21.0102	0.883906
$566$	13.1749	0.553781
$567$	−6.09841	−0.256109
$568$	−3.93297	−0.165024
$569$	−38.7812	−1.62579	−0.812897	−	0.582407i	$-0.802111\pi$
$569$	−38.7812	−1.62579	−0.812897	+	0.582407i	$0.802111\pi$
$570$	0	0
$571$	17.5374	0.733918	0.366959	−	0.930237i	$-0.380399\pi$
$571$	17.5374	0.733918	0.366959	+	0.930237i	$0.380399\pi$
$572$	−8.44563	−0.353130
$573$	−6.48404	−0.270875
$574$	−7.91308	−0.330286
$575$	0.862168	0.0359549
$576$	−2.66514	−0.111048
$577$	18.6225	0.775265	0.387632	−	0.921814i	$-0.373293\pi$
$577$	18.6225	0.775265	0.387632	+	0.921814i	$0.373293\pi$
$578$	−16.9335	−0.704340
$579$	−0.662573	−0.0275356
$580$	4.31772	0.179283
$581$	−10.8657	−0.450784
$582$	−2.50011	−0.103633
$583$	12.1720	0.504112
$584$	1.56026	0.0645639
$585$	18.9284	0.782593
$586$	7.60243	0.314054
$587$	−39.0644	−1.61236	−0.806180	−	0.591670i	$-0.798469\pi$
$587$	−39.0644	−1.61236	−0.806180	+	0.591670i	$0.798469\pi$
$588$	−0.578669	−0.0238639
$589$	0	0
$590$	3.52109	0.144961
$591$	2.44902	0.100739
$592$	9.35193	0.384362
$593$	36.8648	1.51385	0.756927	−	0.653499i	$-0.226700\pi$
$593$	36.8648	1.51385	0.756927	+	0.653499i	$0.226700\pi$
$594$	5.33646	0.218958
$595$	−0.353065	−0.0144743
$596$	−16.4215	−0.672650
$597$	6.43722	0.263458
$598$	1.43090	0.0585139
$599$	−42.4517	−1.73453	−0.867265	−	0.497847i	$-0.834124\pi$
$599$	−42.4517	−1.73453	−0.867265	+	0.497847i	$0.834124\pi$
$600$	1.80898	0.0738512
$601$	−14.2391	−0.580827	−0.290413	−	0.956901i	$-0.593793\pi$
$601$	−14.2391	−0.580827	−0.290413	+	0.956901i	$0.593793\pi$
$602$	2.96944	0.121025
$603$	9.58174	0.390199
$604$	1.43635	0.0584443
$605$	−11.4305	−0.464718
$606$	−6.96620	−0.282983
$607$	44.3153	1.79870	0.899352	−	0.437225i	$-0.144039\pi$
$607$	44.3153	1.79870	0.899352	+	0.437225i	$0.144039\pi$
$608$	0	0
$609$	1.82520	0.0739609
$610$	16.8419	0.681908
$611$	12.2266	0.494634
$612$	−0.687388	−0.0277860
$613$	−8.05804	−0.325461	−0.162731	−	0.986671i	$-0.552030\pi$
$613$	−8.05804	−0.325461	−0.162731	+	0.986671i	$0.552030\pi$
$614$	−21.2034	−0.855702
$615$	−6.26829	−0.252762
$616$	−1.62784	−0.0655876
$617$	44.2259	1.78047	0.890234	−	0.455503i	$-0.150541\pi$
$617$	44.2259	1.78047	0.890234	+	0.455503i	$0.150541\pi$
$618$	−2.95471	−0.118856
$619$	26.7726	1.07608	0.538042	−	0.842918i	$-0.319164\pi$
$619$	26.7726	1.07608	0.538042	+	0.842918i	$0.319164\pi$
$620$	9.66880	0.388308
$621$	−0.904129	−0.0362814
$622$	31.5530	1.26516
$623$	−0.947298	−0.0379527
$624$	3.00228	0.120187
$625$	0.402981	0.0161193
$626$	2.14974	0.0859208
$627$	0	0
$628$	−14.7226	−0.587495
$629$	2.41203	0.0961740
$630$	3.64833	0.145353
$631$	−7.83816	−0.312032	−0.156016	−	0.987755i	$-0.549865\pi$
$631$	−7.83816	−0.312032	−0.156016	+	0.987755i	$0.549865\pi$
$632$	15.7365	0.625965
$633$	−15.2037	−0.604294
$634$	−8.35113	−0.331666
$635$	−17.1800	−0.681768
$636$	−4.32693	−0.171574
$637$	−5.18824	−0.205566
$638$	5.13443	0.203274
$639$	10.4819	0.414659
$640$	1.36891	0.0541107
$641$	8.40692	0.332053	0.166027	−	0.986121i	$-0.446906\pi$
$641$	8.40692	0.332053	0.166027	+	0.986121i	$0.446906\pi$
$642$	2.43873	0.0962492
$643$	20.6578	0.814663	0.407332	−	0.913280i	$-0.366459\pi$
$643$	20.6578	0.814663	0.407332	+	0.913280i	$0.366459\pi$
$644$	0.275797	0.0108679
$645$	2.35222	0.0926185
$646$	0	0
$647$	26.1506	1.02809	0.514043	−	0.857765i	$-0.328147\pi$
$647$	26.1506	1.02809	0.514043	+	0.857765i	$0.328147\pi$
$648$	6.09841	0.239568
$649$	4.18712	0.164359
$650$	16.2190	0.636160
$651$	4.08723	0.160191
$652$	−23.3097	−0.912878
$653$	−26.2233	−1.02620	−0.513099	−	0.858329i	$-0.671503\pi$
$653$	−26.2233	−1.02620	−0.513099	+	0.858329i	$0.671503\pi$
$654$	7.14383	0.279346
$655$	11.2256	0.438621
$656$	7.91308	0.308954
$657$	−4.15831	−0.162231
$658$	2.35659	0.0918695
$659$	35.3589	1.37739	0.688694	−	0.725052i	$-0.258184\pi$
$659$	35.3589	1.37739	0.688694	+	0.725052i	$0.258184\pi$
$660$	−1.28948	−0.0501931
$661$	3.70891	0.144260	0.0721300	−	0.997395i	$-0.477020\pi$
$661$	3.70891	0.144260	0.0721300	+	0.997395i	$0.477020\pi$
$662$	−4.16252	−0.161781
$663$	0.774341	0.0300729
$664$	10.8657	0.421670
$665$	0	0
$666$	−24.9242	−0.965794
$667$	−0.869901	−0.0336827
$668$	−17.9819	−0.695740
$669$	−7.60937	−0.294195
$670$	−4.92150	−0.190134
$671$	20.0276	0.773157
$672$	0.578669	0.0223226
$673$	4.64446	0.179031	0.0895154	−	0.995985i	$-0.471468\pi$
$673$	4.64446	0.179031	0.0895154	+	0.995985i	$0.471468\pi$
$674$	−4.26524	−0.164291
$675$	−10.2481	−0.394450
$676$	13.9179	0.535302
$677$	16.2392	0.624122	0.312061	−	0.950062i	$-0.398981\pi$
$677$	16.2392	0.624122	0.312061	+	0.950062i	$0.398981\pi$
$678$	−8.88152	−0.341093
$679$	−4.32044	−0.165803
$680$	0.353065	0.0135394
$681$	2.75249	0.105476
$682$	11.4977	0.440270
$683$	−12.7728	−0.488736	−0.244368	−	0.969683i	$-0.578581\pi$
$683$	−12.7728	−0.488736	−0.244368	+	0.969683i	$0.578581\pi$
$684$	0	0
$685$	8.99799	0.343796
$686$	−1.00000	−0.0381802
$687$	6.16756	0.235307
$688$	−2.96944	−0.113209
$689$	−38.7945	−1.47795
$690$	0.218471	0.00831703
$691$	17.7594	0.675599	0.337799	−	0.941218i	$-0.390317\pi$
$691$	17.7594	0.675599	0.337799	+	0.941218i	$0.390317\pi$
$692$	14.5104	0.551603
$693$	4.33843	0.164803
$694$	20.2296	0.767905
$695$	−3.53871	−0.134231
$696$	−1.82520	−0.0691841
$697$	2.04093	0.0773056
$698$	7.80705	0.295501
$699$	−0.917465	−0.0347017
$700$	3.12610	0.118155
$701$	15.7559	0.595092	0.297546	−	0.954707i	$-0.403832\pi$
$701$	15.7559	0.595092	0.297546	+	0.954707i	$0.403832\pi$
$702$	−17.0083	−0.641938
$703$	0	0
$704$	1.62784	0.0613516
$705$	1.86676	0.0703062
$706$	−25.5973	−0.963368
$707$	−12.0383	−0.452747
$708$	−1.48845	−0.0559393
$709$	12.1949	0.457989	0.228994	−	0.973428i	$-0.426456\pi$
$709$	12.1949	0.457989	0.228994	+	0.973428i	$0.426456\pi$
$710$	−5.38386	−0.202053
$711$	−41.9400	−1.57287
$712$	0.947298	0.0355015
$713$	−1.94800	−0.0729531
$714$	0.149249	0.00558551
$715$	−11.5613	−0.432367
$716$	22.6246	0.845522
$717$	7.88048	0.294302
$718$	−12.1661	−0.454036
$719$	27.5161	1.02618	0.513088	−	0.858336i	$-0.328501\pi$
$719$	27.5161	1.02618	0.513088	+	0.858336i	$0.328501\pi$
$720$	−3.64833	−0.135965
$721$	−5.10605	−0.190159
$722$	0	0
$723$	−1.45662	−0.0541724
$724$	−26.7353	−0.993608
$725$	−9.86014	−0.366197
$726$	4.83197	0.179331
$727$	0.500852	0.0185756	0.00928778	−	0.999957i	$-0.497044\pi$
$727$	0.500852	0.0185756	0.00928778	+	0.999957i	$0.497044\pi$
$728$	5.18824	0.192289
$729$	−10.5621	−0.391188
$730$	2.13585	0.0790512
$731$	−0.765871	−0.0283268
$732$	−7.11946	−0.263143
$733$	−24.5270	−0.905924	−0.452962	−	0.891530i	$-0.649633\pi$
$733$	−24.5270	−0.905924	−0.452962	+	0.891530i	$0.649633\pi$
$734$	2.68751	0.0991979
$735$	−0.792143	−0.0292186
$736$	−0.275797	−0.0101660
$737$	−5.85243	−0.215577
$738$	−21.0895	−0.776315
$739$	−17.9155	−0.659034	−0.329517	−	0.944150i	$-0.606886\pi$
$739$	−17.9155	−0.659034	−0.329517	+	0.944150i	$0.606886\pi$
$740$	12.8019	0.470607
$741$	0	0
$742$	−7.47738	−0.274503
$743$	−33.3502	−1.22350	−0.611750	−	0.791051i	$-0.709534\pi$
$743$	−33.3502	−1.22350	−0.611750	+	0.791051i	$0.709534\pi$
$744$	−4.08723	−0.149845
$745$	−22.4795	−0.823584
$746$	8.54484	0.312849
$747$	−28.9586	−1.05954
$748$	0.419849	0.0153512
$749$	4.21439	0.153990
$750$	6.43703	0.235047
$751$	40.9525	1.49438	0.747189	−	0.664612i	$-0.231403\pi$
$751$	40.9525	1.49438	0.747189	+	0.664612i	$0.231403\pi$
$752$	−2.35659	−0.0859360
$753$	−9.00483	−0.328154
$754$	−16.3644	−0.595957
$755$	1.96623	0.0715583
$756$	−3.27824	−0.119229
$757$	27.6185	1.00381	0.501906	−	0.864922i	$-0.332632\pi$
$757$	27.6185	1.00381	0.501906	+	0.864922i	$0.332632\pi$
$758$	31.1455	1.13126
$759$	0.259795	0.00942998
$760$	0	0
$761$	−32.9059	−1.19284	−0.596418	−	0.802674i	$-0.703410\pi$
$761$	−32.9059	−1.19284	−0.596418	+	0.802674i	$0.703410\pi$
$762$	7.26240	0.263089
$763$	12.3453	0.446929
$764$	11.2051	0.405386
$765$	−0.940969	−0.0340208
$766$	25.6663	0.927360
$767$	−13.3452	−0.481866
$768$	−0.578669	−0.0208809
$769$	46.4231	1.67406	0.837031	−	0.547156i	$-0.184290\pi$
$769$	46.4231	1.67406	0.837031	+	0.547156i	$0.184290\pi$
$770$	−2.22836	−0.0803045
$771$	−10.0934	−0.363504
$772$	1.14499	0.0412092
$773$	6.11539	0.219955	0.109978	−	0.993934i	$-0.464922\pi$
$773$	6.11539	0.219955	0.109978	+	0.993934i	$0.464922\pi$
$774$	7.91397	0.284462
$775$	−22.0801	−0.793142
$776$	4.32044	0.155095
$777$	5.41167	0.194143
$778$	−13.6211	−0.488339
$779$	0	0
$780$	4.10983	0.147156
$781$	−6.40225	−0.229091
$782$	−0.0711330	−0.00254371
$783$	10.3400	0.369522
$784$	1.00000	0.0357143
$785$	−20.1538	−0.719320
$786$	−4.74534	−0.169261
$787$	−53.2270	−1.89734	−0.948669	−	0.316271i	$-0.897569\pi$
$787$	−53.2270	−1.89734	−0.948669	+	0.316271i	$0.897569\pi$
$788$	−4.23216	−0.150764
$789$	0.404019	0.0143835
$790$	21.5418	0.766422
$791$	−15.3482	−0.545718
$792$	−4.33843	−0.154159
$793$	−63.8318	−2.26673
$794$	−13.8436	−0.491292
$795$	−5.92316	−0.210073
$796$	−11.1242	−0.394286
$797$	49.1901	1.74240	0.871201	−	0.490927i	$-0.163342\pi$
$797$	49.1901	1.74240	0.871201	+	0.490927i	$0.163342\pi$
$798$	0	0
$799$	−0.607807	−0.0215027
$800$	−3.12610	−0.110524
$801$	−2.52468	−0.0892054
$802$	−36.7080	−1.29621
$803$	2.53985	0.0896294
$804$	2.08044	0.0733713
$805$	0.377540	0.0133065
$806$	−36.6454	−1.29078
$807$	−9.32657	−0.328311
$808$	12.0383	0.423506
$809$	−2.51361	−0.0883738	−0.0441869	−	0.999023i	$-0.514070\pi$
$809$	−2.51361	−0.0883738	−0.0441869	+	0.999023i	$0.514070\pi$
$810$	8.34814	0.293324
$811$	33.0435	1.16031	0.580156	−	0.814505i	$-0.302991\pi$
$811$	33.0435	1.16031	0.580156	+	0.814505i	$0.302991\pi$
$812$	−3.15414	−0.110689
$813$	−11.9331	−0.418514
$814$	15.2235	0.533582
$815$	−31.9087	−1.11771
$816$	−0.149249	−0.00522477
$817$	0	0
$818$	−10.2376	−0.357951
$819$	−13.8274	−0.483169
$820$	10.8323	0.378279
$821$	−26.6570	−0.930337	−0.465168	−	0.885222i	$-0.654006\pi$
$821$	−26.6570	−0.930337	−0.465168	+	0.885222i	$0.654006\pi$
$822$	−3.80367	−0.132668
$823$	49.4576	1.72398	0.861992	−	0.506921i	$-0.169216\pi$
$823$	49.4576	1.72398	0.861992	+	0.506921i	$0.169216\pi$
$824$	5.10605	0.177878
$825$	2.94473	0.102522
$826$	−2.57219	−0.0894980
$827$	52.4173	1.82273	0.911365	−	0.411600i	$-0.135030\pi$
$827$	52.4173	1.82273	0.911365	+	0.411600i	$0.135030\pi$
$828$	0.735038	0.0255443
$829$	30.9801	1.07598	0.537992	−	0.842950i	$-0.319183\pi$
$829$	30.9801	1.07598	0.537992	+	0.842950i	$0.319183\pi$
$830$	14.8741	0.516287
$831$	15.3558	0.532686
$832$	−5.18824	−0.179870
$833$	0.257918	0.00893633
$834$	1.49590	0.0517987
$835$	−24.6155	−0.851855
$836$	0	0
$837$	23.1548	0.800346
$838$	16.2221	0.560384
$839$	−28.2511	−0.975338	−0.487669	−	0.873029i	$-0.662153\pi$
$839$	−28.2511	−0.975338	−0.487669	+	0.873029i	$0.662153\pi$
$840$	0.792143	0.0273315
$841$	−19.0514	−0.656945
$842$	11.8248	0.407509
$843$	−4.74588	−0.163457
$844$	26.2736	0.904375
$845$	19.0522	0.655417
$846$	6.28065	0.215933
$847$	8.35013	0.286914
$848$	7.47738	0.256774
$849$	−7.62389	−0.261651
$850$	−0.806277	−0.0276551
$851$	−2.57923	−0.0884150
$852$	2.27589	0.0779707
$853$	−10.7175	−0.366959	−0.183479	−	0.983024i	$-0.558736\pi$
$853$	−10.7175	−0.366959	−0.183479	+	0.983024i	$0.558736\pi$
$854$	−12.3032	−0.421006
$855$	0	0
$856$	−4.21439	−0.144045
$857$	−37.1639	−1.26949	−0.634747	−	0.772720i	$-0.718896\pi$
$857$	−37.1639	−1.26949	−0.634747	+	0.772720i	$0.718896\pi$
$858$	4.88723	0.166847
$859$	−24.5057	−0.836123	−0.418062	−	0.908419i	$-0.637290\pi$
$859$	−24.5057	−0.836123	−0.418062	+	0.908419i	$0.637290\pi$
$860$	−4.06488	−0.138611
$861$	4.57906	0.156054
$862$	37.1721	1.26609
$863$	−42.9929	−1.46350	−0.731748	−	0.681576i	$-0.761295\pi$
$863$	−42.9929	−1.46350	−0.731748	+	0.681576i	$0.761295\pi$
$864$	3.27824	0.111528
$865$	19.8634	0.675375
$866$	−1.18587	−0.0402974
$867$	9.79888	0.332787
$868$	−7.06316	−0.239739
$869$	25.6165	0.868981
$870$	−2.49853	−0.0847081
$871$	18.6528	0.632027
$872$	−12.3453	−0.418064
$873$	−11.5146	−0.389710
$874$	0	0
$875$	11.1239	0.376055
$876$	−0.902873	−0.0305053
$877$	−34.9372	−1.17974	−0.589872	−	0.807496i	$-0.700822\pi$
$877$	−34.9372	−1.17974	−0.589872	+	0.807496i	$0.700822\pi$
$878$	28.6657	0.967420
$879$	−4.39929	−0.148385
$880$	2.22836	0.0751180
$881$	−23.2171	−0.782204	−0.391102	−	0.920347i	$-0.627906\pi$
$881$	−23.2171	−0.782204	−0.391102	+	0.920347i	$0.627906\pi$
$882$	−2.66514	−0.0897400
$883$	27.2224	0.916108	0.458054	−	0.888924i	$-0.348547\pi$
$883$	27.2224	0.916108	0.458054	+	0.888924i	$0.348547\pi$
$884$	−1.33814	−0.0450065
$885$	−2.03754	−0.0684913
$886$	25.2457	0.848147
$887$	−49.2391	−1.65329	−0.826644	−	0.562725i	$-0.809753\pi$
$887$	−49.2391	−1.65329	−0.826644	+	0.562725i	$0.809753\pi$
$888$	−5.41167	−0.181604
$889$	12.5502	0.420919
$890$	1.29676	0.0434676
$891$	9.92724	0.332575
$892$	13.1498	0.440287
$893$	0	0
$894$	9.50261	0.317815
$895$	30.9710	1.03525
$896$	−1.00000	−0.0334077
$897$	−0.828018	−0.0276467
$898$	−10.2395	−0.341695
$899$	22.2782	0.743019
$900$	8.33150	0.277717
$901$	1.92855	0.0642494
$902$	12.8812	0.428898
$903$	−1.71832	−0.0571821
$904$	15.3482	0.510473
$905$	−36.5980	−1.21656
$906$	−0.831172	−0.0276138
$907$	−31.2791	−1.03861	−0.519303	−	0.854590i	$-0.673809\pi$
$907$	−31.2791	−1.03861	−0.519303	+	0.854590i	$0.673809\pi$
$908$	−4.75659	−0.157853
$909$	−32.0838	−1.06415
$910$	7.10221	0.235436
$911$	−4.77182	−0.158098	−0.0790488	−	0.996871i	$-0.525188\pi$
$911$	−4.77182	−0.158098	−0.0790488	+	0.996871i	$0.525188\pi$
$912$	0	0
$913$	17.6876	0.585374
$914$	−14.5779	−0.482195
$915$	−9.74587	−0.322189
$916$	−10.6582	−0.352156
$917$	−8.20043	−0.270802
$918$	0.845518	0.0279062
$919$	49.3170	1.62682	0.813409	−	0.581693i	$-0.197609\pi$
$919$	49.3170	1.62682	0.813409	+	0.581693i	$0.197609\pi$
$920$	−0.377540	−0.0124471
$921$	12.2698	0.404303
$922$	34.8330	1.14716
$923$	20.4052	0.671645
$924$	0.941981	0.0309889
$925$	−29.2351	−0.961243
$926$	−0.367718	−0.0120840
$927$	−13.6084	−0.446957
$928$	3.15414	0.103540
$929$	43.7705	1.43606	0.718031	−	0.696011i	$-0.245043\pi$
$929$	43.7705	1.43606	0.718031	+	0.696011i	$0.245043\pi$
$930$	−5.59504	−0.183468
$931$	0	0
$932$	1.58547	0.0519339
$933$	−18.2588	−0.597765
$934$	−19.4832	−0.637509
$935$	0.574734	0.0187958
$936$	13.8274	0.451963
$937$	−53.3787	−1.74381	−0.871903	−	0.489678i	$-0.837114\pi$
$937$	−53.3787	−1.74381	−0.871903	+	0.489678i	$0.837114\pi$
$938$	3.59521	0.117388
$939$	−1.24399	−0.0405960
$940$	−3.22595	−0.105219
$941$	−35.9659	−1.17245	−0.586227	−	0.810147i	$-0.699387\pi$
$941$	−35.9659	−1.17245	−0.586227	+	0.810147i	$0.699387\pi$
$942$	8.51950	0.277580
$943$	−2.18240	−0.0710688
$944$	2.57219	0.0837177
$945$	−4.48760	−0.145982
$946$	−4.83377	−0.157159
$947$	−36.2742	−1.17875	−0.589377	−	0.807858i	$-0.700627\pi$
$947$	−36.2742	−1.17875	−0.589377	+	0.807858i	$0.700627\pi$
$948$	−9.10623	−0.295757
$949$	−8.09500	−0.262775
$950$	0	0
$951$	4.83254	0.156706
$952$	−0.257918	−0.00835917
$953$	11.1181	0.360151	0.180075	−	0.983653i	$-0.442366\pi$
$953$	11.1181	0.360151	0.180075	+	0.983653i	$0.442366\pi$
$954$	−19.9283	−0.645202
$955$	15.3387	0.496349
$956$	−13.6183	−0.440447
$957$	−2.97114	−0.0960433
$958$	−30.7828	−0.994546
$959$	−6.57313	−0.212257
$960$	−0.792143	−0.0255663
$961$	18.8883	0.609298
$962$	−48.5201	−1.56435
$963$	11.2319	0.361944
$964$	2.51719	0.0810734
$965$	1.56739	0.0504560
$966$	−0.159595	−0.00513489
$967$	51.5662	1.65826	0.829129	−	0.559057i	$-0.188837\pi$
$967$	51.5662	1.65826	0.829129	+	0.559057i	$0.188837\pi$
$968$	−8.35013	−0.268383
$969$	0	0
$970$	5.91428	0.189896
$971$	−13.2758	−0.426042	−0.213021	−	0.977048i	$-0.568330\pi$
$971$	−13.2758	−0.426042	−0.213021	+	0.977048i	$0.568330\pi$
$972$	−13.3637	−0.428641
$973$	2.58506	0.0828734
$974$	−21.5085	−0.689176
$975$	−9.38541	−0.300574
$976$	12.3032	0.393815
$977$	−17.5847	−0.562583	−0.281291	−	0.959622i	$-0.590763\pi$
$977$	−17.5847	−0.562583	−0.281291	+	0.959622i	$0.590763\pi$
$978$	13.4886	0.431318
$979$	1.54205	0.0492842
$980$	1.36891	0.0437281
$981$	32.9019	1.05048
$982$	6.94462	0.221612
$983$	−50.5423	−1.61205	−0.806024	−	0.591883i	$-0.798385\pi$
$983$	−50.5423	−1.61205	−0.806024	+	0.591883i	$0.798385\pi$
$984$	−4.57906	−0.145975
$985$	−5.79343	−0.184594
$986$	0.813509	0.0259074
$987$	−1.36369	−0.0434066
$988$	0	0
$989$	0.818961	0.0260414
$990$	−5.93890	−0.188751
$991$	−45.7977	−1.45481	−0.727406	−	0.686208i	$-0.759274\pi$
$991$	−45.7977	−1.45481	−0.727406	+	0.686208i	$0.759274\pi$
$992$	7.06316	0.224256
$993$	2.40872	0.0764385
$994$	3.93297	0.124746
$995$	−15.2279	−0.482758
$996$	−6.28763	−0.199231
$997$	−14.4959	−0.459088	−0.229544	−	0.973298i	$-0.573724\pi$
$997$	−14.4959	−0.459088	−0.229544	+	0.973298i	$0.573724\pi$
$998$	−16.0613	−0.508412
$999$	30.6579	0.969973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bi.1.3	yes	8
19.18	odd	2		5054.2.a.bf.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bf.1.6	✓	8	19.18	odd	2
5054.2.a.bi.1.3	yes	8	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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.578669 q^{3} +1.00000 q^{4} +1.36891 q^{5} -0.578669 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.66514 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.578669 q^{3} +1.00000 q^{4} +1.36891 q^{5} -0.578669 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.66514 q^{9} +1.36891 q^{10} +1.62784 q^{11} -0.578669 q^{12} -5.18824 q^{13} -1.00000 q^{14} -0.792143 q^{15} +1.00000 q^{16} +0.257918 q^{17} -2.66514 q^{18} +1.36891 q^{20} +0.578669 q^{21} +1.62784 q^{22} -0.275797 q^{23} -0.578669 q^{24} -3.12610 q^{25} -5.18824 q^{26} +3.27824 q^{27} -1.00000 q^{28} +3.15414 q^{29} -0.792143 q^{30} +7.06316 q^{31} +1.00000 q^{32} -0.941981 q^{33} +0.257918 q^{34} -1.36891 q^{35} -2.66514 q^{36} +9.35193 q^{37} +3.00228 q^{39} +1.36891 q^{40} +7.91308 q^{41} +0.578669 q^{42} -2.96944 q^{43} +1.62784 q^{44} -3.64833 q^{45} -0.275797 q^{46} -2.35659 q^{47} -0.578669 q^{48} +1.00000 q^{49} -3.12610 q^{50} -0.149249 q^{51} -5.18824 q^{52} +7.47738 q^{53} +3.27824 q^{54} +2.22836 q^{55} -1.00000 q^{56} +3.15414 q^{58} +2.57219 q^{59} -0.792143 q^{60} +12.3032 q^{61} +7.06316 q^{62} +2.66514 q^{63} +1.00000 q^{64} -7.10221 q^{65} -0.941981 q^{66} -3.59521 q^{67} +0.257918 q^{68} +0.159595 q^{69} -1.36891 q^{70} -3.93297 q^{71} -2.66514 q^{72} +1.56026 q^{73} +9.35193 q^{74} +1.80898 q^{75} -1.62784 q^{77} +3.00228 q^{78} +15.7365 q^{79} +1.36891 q^{80} +6.09841 q^{81} +7.91308 q^{82} +10.8657 q^{83} +0.578669 q^{84} +0.353065 q^{85} -2.96944 q^{86} -1.82520 q^{87} +1.62784 q^{88} +0.947298 q^{89} -3.64833 q^{90} +5.18824 q^{91} -0.275797 q^{92} -4.08723 q^{93} -2.35659 q^{94} -0.578669 q^{96} +4.32044 q^{97} +1.00000 q^{98} -4.33843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} + 8 q^{8} + 8 q^{9} + 6 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 8 q^{14} + 8 q^{15} + 8 q^{16} + 2 q^{17} + 8 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} + 22 q^{27} - 8 q^{28} + 16 q^{29} + 8 q^{30} + 22 q^{31} + 8 q^{32} - 8 q^{33} + 2 q^{34} - 6 q^{35} + 8 q^{36} - 12 q^{37} + 8 q^{39} + 6 q^{40} + 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} + 20 q^{46} + 6 q^{47} + 4 q^{48} + 8 q^{49} - 8 q^{50} - 4 q^{51} + 6 q^{52} + 16 q^{53} + 22 q^{54} + 18 q^{55} - 8 q^{56} + 16 q^{58} + 14 q^{59} + 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} + 32 q^{65} - 8 q^{66} - 20 q^{67} + 2 q^{68} + 40 q^{69} - 6 q^{70} + 8 q^{72} - 18 q^{73} - 12 q^{74} + 16 q^{75} - 4 q^{77} + 8 q^{78} + 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} - 4 q^{84} - 16 q^{85} + 28 q^{87} + 4 q^{88} + 18 q^{89} - 4 q^{90} - 6 q^{91} + 20 q^{92} + 16 q^{93} + 6 q^{94} + 4 q^{96} + 26 q^{97} + 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 + 8 * q^8 + 8 * q^9 + 6 * q^10 + 4 * q^11 + 4 * q^12 + 6 * q^13 - 8 * q^14 + 8 * q^15 + 8 * q^16 + 2 * q^17 + 8 * q^18 + 6 * q^20 - 4 * q^21 + 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 + 22 * q^27 - 8 * q^28 + 16 * q^29 + 8 * q^30 + 22 * q^31 + 8 * q^32 - 8 * q^33 + 2 * q^34 - 6 * q^35 + 8 * q^36 - 12 * q^37 + 8 * q^39 + 6 * q^40 + 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 + 20 * q^46 + 6 * q^47 + 4 * q^48 + 8 * q^49 - 8 * q^50 - 4 * q^51 + 6 * q^52 + 16 * q^53 + 22 * q^54 + 18 * q^55 - 8 * q^56 + 16 * q^58 + 14 * q^59 + 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 + 32 * q^65 - 8 * q^66 - 20 * q^67 + 2 * q^68 + 40 * q^69 - 6 * q^70 + 8 * q^72 - 18 * q^73 - 12 * q^74 + 16 * q^75 - 4 * q^77 + 8 * q^78 + 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 - 4 * q^84 - 16 * q^85 + 28 * q^87 + 4 * q^88 + 18 * q^89 - 4 * q^90 - 6 * q^91 + 20 * q^92 + 16 * q^93 + 6 * q^94 + 4 * q^96 + 26 * q^97 + 8 * q^98 + 4 * q^99

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