LMFDB - Embedded newform 1339.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31499 q^{2} +1.57292 q^{3} +3.35919 q^{4} -1.84365 q^{5} -3.64130 q^{6} +3.43971 q^{7} -3.14652 q^{8} -0.525924 q^{9} +O(q^{10})$	\(q-2.31499 q^{2} +1.57292 q^{3} +3.35919 q^{4} -1.84365 q^{5} -3.64130 q^{6} +3.43971 q^{7} -3.14652 q^{8} -0.525924 q^{9} +4.26804 q^{10} +0.201232 q^{11} +5.28374 q^{12} -1.00000 q^{13} -7.96292 q^{14} -2.89991 q^{15} +0.565791 q^{16} -2.22820 q^{17} +1.21751 q^{18} -5.08976 q^{19} -6.19318 q^{20} +5.41040 q^{21} -0.465850 q^{22} -1.10775 q^{23} -4.94923 q^{24} -1.60095 q^{25} +2.31499 q^{26} -5.54599 q^{27} +11.5547 q^{28} -9.16049 q^{29} +6.71328 q^{30} +9.57554 q^{31} +4.98324 q^{32} +0.316521 q^{33} +5.15827 q^{34} -6.34163 q^{35} -1.76668 q^{36} -5.38729 q^{37} +11.7828 q^{38} -1.57292 q^{39} +5.80109 q^{40} +6.22251 q^{41} -12.5250 q^{42} -1.75193 q^{43} +0.675976 q^{44} +0.969619 q^{45} +2.56443 q^{46} -7.84389 q^{47} +0.889944 q^{48} +4.83164 q^{49} +3.70620 q^{50} -3.50478 q^{51} -3.35919 q^{52} -6.31061 q^{53} +12.8389 q^{54} -0.371001 q^{55} -10.8231 q^{56} -8.00578 q^{57} +21.2065 q^{58} -8.81562 q^{59} -9.74137 q^{60} +5.31677 q^{61} -22.1673 q^{62} -1.80903 q^{63} -12.6678 q^{64} +1.84365 q^{65} -0.732744 q^{66} +15.3374 q^{67} -7.48496 q^{68} -1.74240 q^{69} +14.6808 q^{70} -2.37547 q^{71} +1.65483 q^{72} -5.85486 q^{73} +12.4715 q^{74} -2.51817 q^{75} -17.0975 q^{76} +0.692179 q^{77} +3.64130 q^{78} +8.04086 q^{79} -1.04312 q^{80} -7.14563 q^{81} -14.4051 q^{82} -14.6910 q^{83} +18.1746 q^{84} +4.10803 q^{85} +4.05572 q^{86} -14.4087 q^{87} -0.633180 q^{88} +10.4748 q^{89} -2.24466 q^{90} -3.43971 q^{91} -3.72114 q^{92} +15.0616 q^{93} +18.1586 q^{94} +9.38374 q^{95} +7.83824 q^{96} +10.0674 q^{97} -11.1852 q^{98} -0.105832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.31499	−1.63695	−0.818474	−	0.574544i	$-0.805179\pi$
$2$	−2.31499	−1.63695	−0.818474	+	0.574544i	$0.805179\pi$
$3$	1.57292	0.908126	0.454063	−	0.890970i	$-0.349974\pi$
$3$	1.57292	0.908126	0.454063	+	0.890970i	$0.349974\pi$
$4$	3.35919	1.67960
$5$	−1.84365	−0.824506	−0.412253	−	0.911070i	$-0.635258\pi$
$5$	−1.84365	−0.824506	−0.412253	+	0.911070i	$0.635258\pi$
$6$	−3.64130	−1.48655
$7$	3.43971	1.30009	0.650045	−	0.759896i	$-0.274750\pi$
$7$	3.43971	1.30009	0.650045	+	0.759896i	$0.274750\pi$
$8$	−3.14652	−1.11246
$9$	−0.525924	−0.175308
$10$	4.26804	1.34967
$11$	0.201232	0.0606736	0.0303368	−	0.999540i	$-0.490342\pi$
$11$	0.201232	0.0606736	0.0303368	+	0.999540i	$0.490342\pi$
$12$	5.28374	1.52528
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−7.96292	−2.12818
$15$	−2.89991	−0.748755
$16$	0.565791	0.141448
$17$	−2.22820	−0.540418	−0.270209	−	0.962802i	$-0.587093\pi$
$17$	−2.22820	−0.540418	−0.270209	+	0.962802i	$0.587093\pi$
$18$	1.21751	0.286970
$19$	−5.08976	−1.16767	−0.583836	−	0.811872i	$-0.698449\pi$
$19$	−5.08976	−1.16767	−0.583836	+	0.811872i	$0.698449\pi$
$20$	−6.19318	−1.38484
$21$	5.41040	1.18065
$22$	−0.465850	−0.0993195
$23$	−1.10775	−0.230982	−0.115491	−	0.993309i	$-0.536844\pi$
$23$	−1.10775	−0.230982	−0.115491	+	0.993309i	$0.536844\pi$
$24$	−4.94923	−1.01026
$25$	−1.60095	−0.320191
$26$	2.31499	0.454007
$27$	−5.54599	−1.06733
$28$	11.5547	2.18363
$29$	−9.16049	−1.70106	−0.850530	−	0.525926i	$-0.823719\pi$
$29$	−9.16049	−1.70106	−0.850530	+	0.525926i	$0.823719\pi$
$30$	6.71328	1.22567
$31$	9.57554	1.71982	0.859909	−	0.510447i	$-0.170520\pi$
$31$	9.57554	1.71982	0.859909	+	0.510447i	$0.170520\pi$
$32$	4.98324	0.880921
$33$	0.316521	0.0550993
$34$	5.15827	0.884636
$35$	−6.34163	−1.07193
$36$	−1.76668	−0.294446
$37$	−5.38729	−0.885665	−0.442832	−	0.896604i	$-0.646026\pi$
$37$	−5.38729	−0.885665	−0.442832	+	0.896604i	$0.646026\pi$
$38$	11.7828	1.91142
$39$	−1.57292	−0.251869
$40$	5.80109	0.917232
$41$	6.22251	0.971793	0.485896	−	0.874016i	$-0.338493\pi$
$41$	6.22251	0.971793	0.485896	+	0.874016i	$0.338493\pi$
$42$	−12.5250	−1.93265
$43$	−1.75193	−0.267167	−0.133584	−	0.991038i	$-0.542649\pi$
$43$	−1.75193	−0.267167	−0.133584	+	0.991038i	$0.542649\pi$
$44$	0.675976	0.101907
$45$	0.969619	0.144542
$46$	2.56443	0.378105
$47$	−7.84389	−1.14415	−0.572075	−	0.820202i	$-0.693861\pi$
$47$	−7.84389	−1.14415	−0.572075	+	0.820202i	$0.693861\pi$
$48$	0.889944	0.128452
$49$	4.83164	0.690234
$50$	3.70620	0.524135
$51$	−3.50478	−0.490768
$52$	−3.35919	−0.465836
$53$	−6.31061	−0.866829	−0.433415	−	0.901195i	$-0.642691\pi$
$53$	−6.31061	−0.866829	−0.433415	+	0.901195i	$0.642691\pi$
$54$	12.8389	1.74716
$55$	−0.371001	−0.0500257
$56$	−10.8231	−1.44630
$57$	−8.00578	−1.06039
$58$	21.2065	2.78455
$59$	−8.81562	−1.14770	−0.573848	−	0.818962i	$-0.694550\pi$
$59$	−8.81562	−1.14770	−0.573848	+	0.818962i	$0.694550\pi$
$60$	−9.74137	−1.25761
$61$	5.31677	0.680743	0.340372	−	0.940291i	$-0.389447\pi$
$61$	5.31677	0.680743	0.340372	+	0.940291i	$0.389447\pi$
$62$	−22.1673	−2.81525
$63$	−1.80903	−0.227916
$64$	−12.6678	−1.58347
$65$	1.84365	0.228677
$66$	−0.732744	−0.0901946
$67$	15.3374	1.87376	0.936879	−	0.349653i	$-0.113701\pi$
$67$	15.3374	1.87376	0.936879	+	0.349653i	$0.113701\pi$
$68$	−7.48496	−0.907685
$69$	−1.74240	−0.209760
$70$	14.6808	1.75470
$71$	−2.37547	−0.281917	−0.140958	−	0.990016i	$-0.545018\pi$
$71$	−2.37547	−0.281917	−0.140958	+	0.990016i	$0.545018\pi$
$72$	1.65483	0.195024
$73$	−5.85486	−0.685260	−0.342630	−	0.939470i	$-0.611318\pi$
$73$	−5.85486	−0.685260	−0.342630	+	0.939470i	$0.611318\pi$
$74$	12.4715	1.44979
$75$	−2.51817	−0.290773
$76$	−17.0975	−1.96122
$77$	0.692179	0.0788812
$78$	3.64130	0.412296
$79$	8.04086	0.904667	0.452333	−	0.891849i	$-0.350592\pi$
$79$	8.04086	0.904667	0.452333	+	0.891849i	$0.350592\pi$
$80$	−1.04312	−0.116625
$81$	−7.14563	−0.793959
$82$	−14.4051	−1.59077
$83$	−14.6910	−1.61254	−0.806271	−	0.591546i	$-0.798518\pi$
$83$	−14.6910	−1.61254	−0.806271	+	0.591546i	$0.798518\pi$
$84$	18.1746	1.98301
$85$	4.10803	0.445578
$86$	4.05572	0.437339
$87$	−14.4087	−1.54478
$88$	−0.633180	−0.0674972
$89$	10.4748	1.11033	0.555164	−	0.831741i	$-0.312655\pi$
$89$	10.4748	1.11033	0.555164	+	0.831741i	$0.312655\pi$
$90$	−2.24466	−0.236608
$91$	−3.43971	−0.360580
$92$	−3.72114	−0.387956
$93$	15.0616	1.56181
$94$	18.1586	1.87291
$95$	9.38374	0.962751
$96$	7.83824	0.799987
$97$	10.0674	1.02219	0.511094	−	0.859525i	$-0.329240\pi$
$97$	10.0674	1.02219	0.511094	+	0.859525i	$0.329240\pi$
$98$	−11.1852	−1.12988
$99$	−0.105832	−0.0106366
$100$	−5.37791	−0.537791
$101$	0.964403	0.0959617	0.0479809	−	0.998848i	$-0.484721\pi$
$101$	0.964403	0.0959617	0.0479809	+	0.998848i	$0.484721\pi$
$102$	8.11355	0.803361
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	3.14652	0.308542
$105$	−9.97488	−0.973448
$106$	14.6090	1.41895
$107$	−5.52274	−0.533904	−0.266952	−	0.963710i	$-0.586016\pi$
$107$	−5.52274	−0.533904	−0.266952	+	0.963710i	$0.586016\pi$
$108$	−18.6301	−1.79268
$109$	−3.05619	−0.292730	−0.146365	−	0.989231i	$-0.546757\pi$
$109$	−3.05619	−0.292730	−0.146365	+	0.989231i	$0.546757\pi$
$110$	0.858864	0.0818895
$111$	−8.47377	−0.804295
$112$	1.94616	0.183895
$113$	3.19343	0.300413	0.150207	−	0.988655i	$-0.452006\pi$
$113$	3.19343	0.300413	0.150207	+	0.988655i	$0.452006\pi$
$114$	18.5333	1.73581
$115$	2.04230	0.190446
$116$	−30.7719	−2.85710
$117$	0.525924	0.0486217
$118$	20.4081	1.87872
$119$	−7.66438	−0.702593
$120$	9.12464	0.832962
$121$	−10.9595	−0.996319
$122$	−12.3083	−1.11434
$123$	9.78751	0.882510
$124$	32.1661	2.88860
$125$	12.1699	1.08850
$126$	4.18789	0.373086
$127$	2.62200	0.232665	0.116332	−	0.993210i	$-0.462886\pi$
$127$	2.62200	0.232665	0.116332	+	0.993210i	$0.462886\pi$
$128$	19.3593	1.71113
$129$	−2.75565	−0.242622
$130$	−4.26804	−0.374332
$131$	−12.9844	−1.13445	−0.567227	−	0.823561i	$-0.691984\pi$
$131$	−12.9844	−1.13445	−0.567227	+	0.823561i	$0.691984\pi$
$132$	1.06326	0.0925445
$133$	−17.5073	−1.51808
$134$	−35.5059	−3.06724
$135$	10.2249	0.880017
$136$	7.01109	0.601196
$137$	−6.72502	−0.574557	−0.287279	−	0.957847i	$-0.592751\pi$
$137$	−6.72502	−0.574557	−0.287279	+	0.957847i	$0.592751\pi$
$138$	4.03364	0.343366
$139$	−18.9429	−1.60672	−0.803358	−	0.595497i	$-0.796955\pi$
$139$	−18.9429	−1.60672	−0.803358	+	0.595497i	$0.796955\pi$
$140$	−21.3028	−1.80041
$141$	−12.3378	−1.03903
$142$	5.49920	0.461483
$143$	−0.201232	−0.0168278
$144$	−0.297563	−0.0247969
$145$	16.8887	1.40253
$146$	13.5540	1.12173
$147$	7.59978	0.626819
$148$	−18.0969	−1.48756
$149$	−15.0219	−1.23064	−0.615320	−	0.788277i	$-0.710973\pi$
$149$	−15.0219	−1.23064	−0.615320	+	0.788277i	$0.710973\pi$
$150$	5.82955	0.475981
$151$	9.55246	0.777368	0.388684	−	0.921371i	$-0.372930\pi$
$151$	9.55246	0.777368	0.388684	+	0.921371i	$0.372930\pi$
$152$	16.0150	1.29899
$153$	1.17186	0.0947396
$154$	−1.60239	−0.129124
$155$	−17.6540	−1.41800
$156$	−5.28374	−0.423038
$157$	−2.52321	−0.201374	−0.100687	−	0.994918i	$-0.532104\pi$
$157$	−2.52321	−0.201374	−0.100687	+	0.994918i	$0.532104\pi$
$158$	−18.6145	−1.48089
$159$	−9.92609	−0.787190
$160$	−9.18736	−0.726324
$161$	−3.81034	−0.300297
$162$	16.5421	1.29967
$163$	−14.7405	−1.15457	−0.577284	−	0.816543i	$-0.695888\pi$
$163$	−14.7405	−1.15457	−0.577284	+	0.816543i	$0.695888\pi$
$164$	20.9026	1.63222
$165$	−0.583554	−0.0454296
$166$	34.0095	2.63965
$167$	−3.47508	−0.268910	−0.134455	−	0.990920i	$-0.542928\pi$
$167$	−3.47508	−0.268910	−0.134455	+	0.990920i	$0.542928\pi$
$168$	−17.0239	−1.31342
$169$	1.00000	0.0769231
$170$	−9.51005	−0.729388
$171$	2.67683	0.204702
$172$	−5.88508	−0.448734
$173$	7.94069	0.603719	0.301860	−	0.953352i	$-0.402393\pi$
$173$	7.94069	0.603719	0.301860	+	0.953352i	$0.402393\pi$
$174$	33.3561	2.52872
$175$	−5.50682	−0.416277
$176$	0.113855	0.00858215
$177$	−13.8663	−1.04225
$178$	−24.2491	−1.81755
$179$	11.5509	0.863357	0.431679	−	0.902027i	$-0.357921\pi$
$179$	11.5509	0.863357	0.431679	+	0.902027i	$0.357921\pi$
$180$	3.25714	0.242773
$181$	−5.27852	−0.392349	−0.196175	−	0.980569i	$-0.562852\pi$
$181$	−5.27852	−0.392349	−0.196175	+	0.980569i	$0.562852\pi$
$182$	7.96292	0.590251
$183$	8.36286	0.618200
$184$	3.48556	0.256959
$185$	9.93228	0.730235
$186$	−34.8674	−2.55660
$187$	−0.448385	−0.0327891
$188$	−26.3491	−1.92171
$189$	−19.0766	−1.38762
$190$	−21.7233	−1.57597
$191$	−16.2797	−1.17796	−0.588978	−	0.808149i	$-0.700469\pi$
$191$	−16.2797	−1.17796	−0.588978	+	0.808149i	$0.700469\pi$
$192$	−19.9254	−1.43799
$193$	−18.2437	−1.31321	−0.656607	−	0.754233i	$-0.728009\pi$
$193$	−18.2437	−1.31321	−0.656607	+	0.754233i	$0.728009\pi$
$194$	−23.3059	−1.67327
$195$	2.89991	0.207667
$196$	16.2304	1.15931
$197$	12.3575	0.880437	0.440218	−	0.897891i	$-0.354901\pi$
$197$	12.3575	0.880437	0.440218	+	0.897891i	$0.354901\pi$
$198$	0.245001	0.0174115
$199$	−16.7216	−1.18536	−0.592682	−	0.805437i	$-0.701931\pi$
$199$	−16.7216	−1.18536	−0.592682	+	0.805437i	$0.701931\pi$
$200$	5.03744	0.356200
$201$	24.1245	1.70161
$202$	−2.23259	−0.157084
$203$	−31.5095	−2.21153
$204$	−11.7732	−0.824292
$205$	−11.4721	−0.801249
$206$	2.31499	0.161293
$207$	0.582591	0.0404929
$208$	−0.565791	−0.0392306
$209$	−1.02422	−0.0708468
$210$	23.0918	1.59348
$211$	−26.6463	−1.83440	−0.917202	−	0.398423i	$-0.869558\pi$
$211$	−26.6463	−1.83440	−0.917202	+	0.398423i	$0.869558\pi$
$212$	−21.1986	−1.45592
$213$	−3.73643	−0.256016
$214$	12.7851	0.873973
$215$	3.22995	0.220281
$216$	17.4506	1.18736
$217$	32.9371	2.23592
$218$	7.07507	0.479184
$219$	−9.20923	−0.622302
$220$	−1.24626	−0.0840230
$221$	2.22820	0.149885
$222$	19.6167	1.31659
$223$	−27.7264	−1.85670	−0.928350	−	0.371708i	$-0.878772\pi$
$223$	−27.7264	−1.85670	−0.928350	+	0.371708i	$0.878772\pi$
$224$	17.1409	1.14528
$225$	0.841979	0.0561319
$226$	−7.39278	−0.491760
$227$	19.0977	1.26756	0.633780	−	0.773513i	$-0.281502\pi$
$227$	19.0977	1.26756	0.633780	+	0.773513i	$0.281502\pi$
$228$	−26.8930	−1.78103
$229$	−9.12301	−0.602865	−0.301433	−	0.953487i	$-0.597465\pi$
$229$	−9.12301	−0.602865	−0.301433	+	0.953487i	$0.597465\pi$
$230$	−4.72791	−0.311749
$231$	1.08874	0.0716340
$232$	28.8237	1.89237
$233$	22.0872	1.44698	0.723491	−	0.690334i	$-0.242536\pi$
$233$	22.0872	1.44698	0.723491	+	0.690334i	$0.242536\pi$
$234$	−1.21751	−0.0795911
$235$	14.4614	0.943357
$236$	−29.6134	−1.92767
$237$	12.6476	0.821551
$238$	17.7430	1.15011
$239$	20.9677	1.35629	0.678143	−	0.734930i	$-0.262785\pi$
$239$	20.9677	1.35629	0.678143	+	0.734930i	$0.262785\pi$
$240$	−1.64075	−0.105910
$241$	11.6465	0.750219	0.375110	−	0.926980i	$-0.377605\pi$
$241$	11.6465	0.750219	0.375110	+	0.926980i	$0.377605\pi$
$242$	25.3712	1.63092
$243$	5.39848	0.346312
$244$	17.8601	1.14337
$245$	−8.90785	−0.569102
$246$	−22.6580	−1.44462
$247$	5.08976	0.323854
$248$	−30.1297	−1.91324
$249$	−23.1077	−1.46439
$250$	−28.1731	−1.78182
$251$	−17.2949	−1.09164	−0.545822	−	0.837901i	$-0.683783\pi$
$251$	−17.2949	−1.09164	−0.545822	+	0.837901i	$0.683783\pi$
$252$	−6.07687	−0.382807
$253$	−0.222914	−0.0140145
$254$	−6.06991	−0.380860
$255$	6.46159	0.404641
$256$	−19.4811	−1.21757
$257$	31.1894	1.94554	0.972770	−	0.231772i	$-0.0744523\pi$
$257$	31.1894	1.94554	0.972770	+	0.231772i	$0.0744523\pi$
$258$	6.37931	0.397159
$259$	−18.5307	−1.15144
$260$	6.19318	0.384085
$261$	4.81772	0.298209
$262$	30.0589	1.85704
$263$	−22.9681	−1.41627	−0.708137	−	0.706075i	$-0.750464\pi$
$263$	−22.9681	−1.41627	−0.708137	+	0.706075i	$0.750464\pi$
$264$	−0.995941	−0.0612959
$265$	11.6346	0.714706
$266$	40.5293	2.48501
$267$	16.4761	1.00832
$268$	51.5212	3.14716
$269$	2.51178	0.153146	0.0765730	−	0.997064i	$-0.475602\pi$
$269$	2.51178	0.153146	0.0765730	+	0.997064i	$0.475602\pi$
$270$	−23.6705	−1.44054
$271$	29.2419	1.77632	0.888161	−	0.459533i	$-0.151983\pi$
$271$	29.2419	1.77632	0.888161	+	0.459533i	$0.151983\pi$
$272$	−1.26070	−0.0764410
$273$	−5.41040	−0.327452
$274$	15.5684	0.940520
$275$	−0.322162	−0.0194271
$276$	−5.85306	−0.352313
$277$	21.2417	1.27629	0.638146	−	0.769916i	$-0.279702\pi$
$277$	21.2417	1.27629	0.638146	+	0.769916i	$0.279702\pi$
$278$	43.8527	2.63011
$279$	−5.03601	−0.301498
$280$	19.9541	1.19248
$281$	−15.2237	−0.908167	−0.454084	−	0.890959i	$-0.650033\pi$
$281$	−15.2237	−0.908167	−0.454084	+	0.890959i	$0.650033\pi$
$282$	28.5619	1.70084
$283$	19.7680	1.17508	0.587541	−	0.809194i	$-0.300096\pi$
$283$	19.7680	1.17508	0.587541	+	0.809194i	$0.300096\pi$
$284$	−7.97967	−0.473506
$285$	14.7599	0.874299
$286$	0.465850	0.0275463
$287$	21.4037	1.26342
$288$	−2.62080	−0.154432
$289$	−12.0351	−0.707948
$290$	−39.0973	−2.29587
$291$	15.8352	0.928276
$292$	−19.6676	−1.15096
$293$	−9.16839	−0.535623	−0.267812	−	0.963471i	$-0.586300\pi$
$293$	−9.16839	−0.535623	−0.267812	+	0.963471i	$0.586300\pi$
$294$	−17.5934	−1.02607
$295$	16.2529	0.946282
$296$	16.9512	0.985270
$297$	−1.11603	−0.0647586
$298$	34.7756	2.01449
$299$	1.10775	0.0640627
$300$	−8.45902	−0.488382
$301$	−6.02615	−0.347342
$302$	−22.1139	−1.27251
$303$	1.51693	0.0871453
$304$	−2.87974	−0.165165
$305$	−9.80227	−0.561277
$306$	−2.71286	−0.155084
$307$	−2.37032	−0.135281	−0.0676406	−	0.997710i	$-0.521547\pi$
$307$	−2.37032	−0.135281	−0.0676406	+	0.997710i	$0.521547\pi$
$308$	2.32516	0.132489
$309$	−1.57292	−0.0894803
$310$	40.8688	2.32119
$311$	30.6968	1.74066	0.870329	−	0.492472i	$-0.163906\pi$
$311$	30.6968	1.74066	0.870329	+	0.492472i	$0.163906\pi$
$312$	4.94923	0.280195
$313$	15.3334	0.866694	0.433347	−	0.901227i	$-0.357332\pi$
$313$	15.3334	0.866694	0.433347	+	0.901227i	$0.357332\pi$
$314$	5.84122	0.329639
$315$	3.33521	0.187918
$316$	27.0108	1.51948
$317$	1.07072	0.0601374	0.0300687	−	0.999548i	$-0.490427\pi$
$317$	1.07072	0.0601374	0.0300687	+	0.999548i	$0.490427\pi$
$318$	22.9788	1.28859
$319$	−1.84338	−0.103209
$320$	23.3549	1.30558
$321$	−8.68683	−0.484852
$322$	8.82091	0.491570
$323$	11.3410	0.631031
$324$	−24.0036	−1.33353
$325$	1.60095	0.0888049
$326$	34.1243	1.88997
$327$	−4.80715	−0.265836
$328$	−19.5793	−1.08108
$329$	−26.9807	−1.48750
$330$	1.35092	0.0743659
$331$	−4.33004	−0.238000	−0.119000	−	0.992894i	$-0.537969\pi$
$331$	−4.33004	−0.238000	−0.119000	+	0.992894i	$0.537969\pi$
$332$	−49.3498	−2.70842
$333$	2.83330	0.155264
$334$	8.04478	0.440191
$335$	−28.2768	−1.54492
$336$	3.06115	0.167000
$337$	18.1982	0.991320	0.495660	−	0.868517i	$-0.334926\pi$
$337$	18.1982	0.991320	0.495660	+	0.868517i	$0.334926\pi$
$338$	−2.31499	−0.125919
$339$	5.02302	0.272813
$340$	13.7997	0.748391
$341$	1.92690	0.104348
$342$	−6.19683	−0.335086
$343$	−7.45855	−0.402724
$344$	5.51250	0.297214
$345$	3.21238	0.172948
$346$	−18.3826	−0.988257
$347$	16.0947	0.864011	0.432006	−	0.901871i	$-0.357806\pi$
$347$	16.0947	0.864011	0.432006	+	0.901871i	$0.357806\pi$
$348$	−48.4017	−2.59460
$349$	14.9702	0.801335	0.400667	−	0.916224i	$-0.368778\pi$
$349$	14.9702	0.801335	0.400667	+	0.916224i	$0.368778\pi$
$350$	12.7483	0.681423
$351$	5.54599	0.296023
$352$	1.00279	0.0534487
$353$	17.8209	0.948512	0.474256	−	0.880387i	$-0.342717\pi$
$353$	17.8209	0.948512	0.474256	+	0.880387i	$0.342717\pi$
$354$	32.1003	1.70611
$355$	4.37954	0.232442
$356$	35.1869	1.86490
$357$	−12.0555	−0.638042
$358$	−26.7403	−1.41327
$359$	7.22502	0.381322	0.190661	−	0.981656i	$-0.438937\pi$
$359$	7.22502	0.381322	0.190661	+	0.981656i	$0.438937\pi$
$360$	−3.05093	−0.160798
$361$	6.90566	0.363456
$362$	12.2197	0.642255
$363$	−17.2384	−0.904783
$364$	−11.5547	−0.605629
$365$	10.7943	0.565001
$366$	−19.3600	−1.01196
$367$	−24.0373	−1.25474	−0.627369	−	0.778722i	$-0.715868\pi$
$367$	−24.0373	−1.25474	−0.627369	+	0.778722i	$0.715868\pi$
$368$	−0.626754	−0.0326718
$369$	−3.27257	−0.170363
$370$	−22.9932	−1.19536
$371$	−21.7067	−1.12696
$372$	50.5947	2.62321
$373$	5.55433	0.287592	0.143796	−	0.989607i	$-0.454069\pi$
$373$	5.55433	0.287592	0.143796	+	0.989607i	$0.454069\pi$
$374$	1.03801	0.0536741
$375$	19.1422	0.988499
$376$	24.6810	1.27282
$377$	9.16049	0.471789
$378$	44.1623	2.27146
$379$	−31.4541	−1.61569	−0.807844	−	0.589396i	$-0.799366\pi$
$379$	−31.4541	−1.61569	−0.807844	+	0.589396i	$0.799366\pi$
$380$	31.5218	1.61703
$381$	4.12419	0.211289
$382$	37.6873	1.92825
$383$	2.45369	0.125378	0.0626888	−	0.998033i	$-0.480032\pi$
$383$	2.45369	0.125378	0.0626888	+	0.998033i	$0.480032\pi$
$384$	30.4506	1.55392
$385$	−1.27614	−0.0650379
$386$	42.2341	2.14966
$387$	0.921383	0.0468366
$388$	33.8183	1.71686
$389$	9.99341	0.506686	0.253343	−	0.967377i	$-0.418470\pi$
$389$	9.99341	0.506686	0.253343	+	0.967377i	$0.418470\pi$
$390$	−6.71328	−0.339940
$391$	2.46829	0.124827
$392$	−15.2029	−0.767860
$393$	−20.4235	−1.03023
$394$	−28.6076	−1.44123
$395$	−14.8245	−0.745903
$396$	−0.355512	−0.0178651
$397$	−10.4883	−0.526392	−0.263196	−	0.964742i	$-0.584777\pi$
$397$	−10.4883	−0.526392	−0.263196	+	0.964742i	$0.584777\pi$
$398$	38.7104	1.94038
$399$	−27.5376	−1.37861
$400$	−0.905805	−0.0452903
$401$	7.58917	0.378985	0.189493	−	0.981882i	$-0.439316\pi$
$401$	7.58917	0.378985	0.189493	+	0.981882i	$0.439316\pi$
$402$	−55.8479	−2.78544
$403$	−9.57554	−0.476992
$404$	3.23962	0.161177
$405$	13.1741	0.654624
$406$	72.9442	3.62016
$407$	−1.08409	−0.0537365
$408$	11.0279	0.545961
$409$	−33.8264	−1.67261	−0.836304	−	0.548266i	$-0.815288\pi$
$409$	−33.8264	−1.67261	−0.836304	+	0.548266i	$0.815288\pi$
$410$	26.5579	1.31160
$411$	−10.5779	−0.521770
$412$	−3.35919	−0.165496
$413$	−30.3232	−1.49211
$414$	−1.34869	−0.0662847
$415$	27.0850	1.32955
$416$	−4.98324	−0.244324
$417$	−29.7957	−1.45910
$418$	2.37106	0.115973
$419$	14.7050	0.718388	0.359194	−	0.933263i	$-0.383052\pi$
$419$	14.7050	0.718388	0.359194	+	0.933263i	$0.383052\pi$
$420$	−33.5075	−1.63500
$421$	1.72052	0.0838532	0.0419266	−	0.999121i	$-0.486650\pi$
$421$	1.72052	0.0838532	0.0419266	+	0.999121i	$0.486650\pi$
$422$	61.6859	3.00282
$423$	4.12529	0.200578
$424$	19.8565	0.964316
$425$	3.56725	0.173037
$426$	8.64980	0.419084
$427$	18.2882	0.885027
$428$	−18.5520	−0.896743
$429$	−0.316521	−0.0152818
$430$	−7.47732	−0.360588
$431$	−6.99409	−0.336893	−0.168447	−	0.985711i	$-0.553875\pi$
$431$	−6.99409	−0.336893	−0.168447	+	0.985711i	$0.553875\pi$
$432$	−3.13788	−0.150971
$433$	−11.5637	−0.555714	−0.277857	−	0.960622i	$-0.589624\pi$
$433$	−11.5637	−0.555714	−0.277857	+	0.960622i	$0.589624\pi$
$434$	−76.2493	−3.66008
$435$	26.5646	1.27368
$436$	−10.2663	−0.491669
$437$	5.63817	0.269710
$438$	21.3193	1.01868
$439$	−7.75413	−0.370084	−0.185042	−	0.982731i	$-0.559242\pi$
$439$	−7.75413	−0.370084	−0.185042	+	0.982731i	$0.559242\pi$
$440$	1.16736	0.0556518
$441$	−2.54107	−0.121003
$442$	−5.15827	−0.245354
$443$	39.9378	1.89750	0.948750	−	0.316028i	$-0.102349\pi$
$443$	39.9378	1.89750	0.948750	+	0.316028i	$0.102349\pi$
$444$	−28.4650	−1.35089
$445$	−19.3119	−0.915472
$446$	64.1865	3.03932
$447$	−23.6282	−1.11758
$448$	−43.5735	−2.05865
$449$	29.3010	1.38280	0.691400	−	0.722472i	$-0.256994\pi$
$449$	29.3010	1.38280	0.691400	+	0.722472i	$0.256994\pi$
$450$	−1.94918	−0.0918850
$451$	1.25217	0.0589622
$452$	10.7274	0.504573
$453$	15.0252	0.705948
$454$	−44.2111	−2.07493
$455$	6.34163	0.297300
$456$	25.1904	1.17965
$457$	29.3954	1.37506	0.687529	−	0.726157i	$-0.258695\pi$
$457$	29.3954	1.37506	0.687529	+	0.726157i	$0.258695\pi$
$458$	21.1197	0.986858
$459$	12.3576	0.576803
$460$	6.86048	0.319872
$461$	−2.30571	−0.107388	−0.0536939	−	0.998557i	$-0.517100\pi$
$461$	−2.30571	−0.107388	−0.0536939	+	0.998557i	$0.517100\pi$
$462$	−2.52043	−0.117261
$463$	−12.0516	−0.560085	−0.280043	−	0.959988i	$-0.590349\pi$
$463$	−12.0516	−0.560085	−0.280043	+	0.959988i	$0.590349\pi$
$464$	−5.18293	−0.240611
$465$	−27.7683	−1.28772
$466$	−51.1318	−2.36863
$467$	11.5208	0.533118	0.266559	−	0.963819i	$-0.414113\pi$
$467$	11.5208	0.533118	0.266559	+	0.963819i	$0.414113\pi$
$468$	1.76668	0.0816648
$469$	52.7562	2.43605
$470$	−33.4780	−1.54423
$471$	−3.96881	−0.182873
$472$	27.7386	1.27677
$473$	−0.352545	−0.0162100
$474$	−29.2792	−1.34484
$475$	8.14847	0.373877
$476$	−25.7461	−1.18007
$477$	3.31890	0.151962
$478$	−48.5400	−2.22017
$479$	14.5002	0.662529	0.331265	−	0.943538i	$-0.392525\pi$
$479$	14.5002	0.662529	0.331265	+	0.943538i	$0.392525\pi$
$480$	−14.4510	−0.659594
$481$	5.38729	0.245639
$482$	−26.9616	−1.22807
$483$	−5.99336	−0.272707
$484$	−36.8151	−1.67341
$485$	−18.5608	−0.842800
$486$	−12.4974	−0.566895
$487$	26.7291	1.21121	0.605605	−	0.795765i	$-0.292931\pi$
$487$	26.7291	1.21121	0.605605	+	0.795765i	$0.292931\pi$
$488$	−16.7293	−0.757302
$489$	−23.1857	−1.04849
$490$	20.6216	0.931590
$491$	−10.3327	−0.466306	−0.233153	−	0.972440i	$-0.574904\pi$
$491$	−10.3327	−0.466306	−0.233153	+	0.972440i	$0.574904\pi$
$492$	32.8781	1.48226
$493$	20.4114	0.919284
$494$	−11.7828	−0.530131
$495$	0.195118	0.00876990
$496$	5.41776	0.243265
$497$	−8.17094	−0.366517
$498$	53.4942	2.39713
$499$	−20.0436	−0.897274	−0.448637	−	0.893714i	$-0.648090\pi$
$499$	−20.0436	−0.897274	−0.448637	+	0.893714i	$0.648090\pi$
$500$	40.8809	1.82825
$501$	−5.46602	−0.244204
$502$	40.0376	1.78696
$503$	−13.0327	−0.581100	−0.290550	−	0.956860i	$-0.593838\pi$
$503$	−13.0327	−0.581100	−0.290550	+	0.956860i	$0.593838\pi$
$504$	5.69214	0.253548
$505$	−1.77802	−0.0791210
$506$	0.516044	0.0229410
$507$	1.57292	0.0698558
$508$	8.80780	0.390783
$509$	12.1005	0.536344	0.268172	−	0.963371i	$-0.413580\pi$
$509$	12.1005	0.536344	0.268172	+	0.963371i	$0.413580\pi$
$510$	−14.9585	−0.662376
$511$	−20.1391	−0.890900
$512$	6.38002	0.281960
$513$	28.2278	1.24629
$514$	−72.2032	−3.18475
$515$	1.84365	0.0812409
$516$	−9.25677	−0.407506
$517$	−1.57844	−0.0694197
$518$	42.8985	1.88485
$519$	12.4901	0.548253
$520$	−5.80109	−0.254394
$521$	−23.4154	−1.02585	−0.512925	−	0.858434i	$-0.671438\pi$
$521$	−23.4154	−1.02585	−0.512925	+	0.858434i	$0.671438\pi$
$522$	−11.1530	−0.488153
$523$	14.1886	0.620422	0.310211	−	0.950668i	$-0.399600\pi$
$523$	14.1886	0.620422	0.310211	+	0.950668i	$0.399600\pi$
$524$	−43.6172	−1.90543
$525$	−8.66179	−0.378032
$526$	53.1710	2.31837
$527$	−21.3362	−0.929422
$528$	0.179085	0.00779367
$529$	−21.7729	−0.946648
$530$	−26.9339	−1.16994
$531$	4.63634	0.201200
$532$	−58.8105	−2.54976
$533$	−6.22251	−0.269527
$534$	−38.1419	−1.65056
$535$	10.1820	0.440207
$536$	−48.2594	−2.08449
$537$	18.1687	0.784037
$538$	−5.81476	−0.250692
$539$	0.972278	0.0418790
$540$	34.3473	1.47807
$541$	−2.03307	−0.0874084	−0.0437042	−	0.999045i	$-0.513916\pi$
$541$	−2.03307	−0.0874084	−0.0437042	+	0.999045i	$0.513916\pi$
$542$	−67.6949	−2.90774
$543$	−8.30269	−0.356302
$544$	−11.1037	−0.476066
$545$	5.63455	0.241358
$546$	12.5250	0.536022
$547$	21.5482	0.921336	0.460668	−	0.887572i	$-0.347610\pi$
$547$	21.5482	0.921336	0.460668	+	0.887572i	$0.347610\pi$
$548$	−22.5906	−0.965024
$549$	−2.79622	−0.119340
$550$	0.745804	0.0318012
$551$	46.6247	1.98628
$552$	5.48250	0.233351
$553$	27.6583	1.17615
$554$	−49.1744	−2.08922
$555$	15.6227	0.663146
$556$	−63.6328	−2.69863
$557$	11.0214	0.466993	0.233496	−	0.972358i	$-0.424983\pi$
$557$	11.0214	0.466993	0.233496	+	0.972358i	$0.424983\pi$
$558$	11.6583	0.493536
$559$	1.75193	0.0740989
$560$	−3.58804	−0.151622
$561$	−0.705273	−0.0297767
$562$	35.2427	1.48662
$563$	−19.5913	−0.825674	−0.412837	−	0.910805i	$-0.635462\pi$
$563$	−19.5913	−0.825674	−0.412837	+	0.910805i	$0.635462\pi$
$564$	−41.4451	−1.74515
$565$	−5.88758	−0.247692
$566$	−45.7627	−1.92355
$567$	−24.5789	−1.03222
$568$	7.47447	0.313622
$569$	−18.7336	−0.785352	−0.392676	−	0.919677i	$-0.628451\pi$
$569$	−18.7336	−0.785352	−0.392676	+	0.919677i	$0.628451\pi$
$570$	−34.1690	−1.43118
$571$	−0.580567	−0.0242960	−0.0121480	−	0.999926i	$-0.503867\pi$
$571$	−0.580567	−0.0242960	−0.0121480	+	0.999926i	$0.503867\pi$
$572$	−0.675976	−0.0282640
$573$	−25.6066	−1.06973
$574$	−49.5493	−2.06815
$575$	1.77345	0.0739581
$576$	6.66227	0.277595
$577$	13.8723	0.577513	0.288757	−	0.957403i	$-0.406758\pi$
$577$	13.8723	0.577513	0.288757	+	0.957403i	$0.406758\pi$
$578$	27.8612	1.15887
$579$	−28.6960	−1.19256
$580$	56.7325	2.35569
$581$	−50.5327	−2.09645
$582$	−36.6584	−1.51954
$583$	−1.26989	−0.0525937
$584$	18.4225	0.762327
$585$	−0.969619	−0.0400888
$586$	21.2248	0.876787
$587$	22.8571	0.943415	0.471708	−	0.881755i	$-0.343638\pi$
$587$	22.8571	0.943415	0.471708	+	0.881755i	$0.343638\pi$
$588$	25.5291	1.05280
$589$	−48.7372	−2.00818
$590$	−37.6254	−1.54901
$591$	19.4374	0.799547
$592$	−3.04808	−0.125275
$593$	−10.7454	−0.441260	−0.220630	−	0.975358i	$-0.570811\pi$
$593$	−10.7454	−0.441260	−0.220630	+	0.975358i	$0.570811\pi$
$594$	2.58360	0.106006
$595$	14.1304	0.579291
$596$	−50.4614	−2.06698
$597$	−26.3018	−1.07646
$598$	−2.56443	−0.104867
$599$	−12.8513	−0.525088	−0.262544	−	0.964920i	$-0.584561\pi$
$599$	−12.8513	−0.525088	−0.262544	+	0.964920i	$0.584561\pi$
$600$	7.92348	0.323475
$601$	4.56545	0.186229	0.0931143	−	0.995655i	$-0.470318\pi$
$601$	4.56545	0.186229	0.0931143	+	0.995655i	$0.470318\pi$
$602$	13.9505	0.568580
$603$	−8.06629	−0.328485
$604$	32.0885	1.30566
$605$	20.2055	0.821470
$606$	−3.51168	−0.142652
$607$	−39.1903	−1.59069	−0.795343	−	0.606160i	$-0.792709\pi$
$607$	−39.1903	−1.59069	−0.795343	+	0.606160i	$0.792709\pi$
$608$	−25.3635	−1.02863
$609$	−49.5619	−2.00835
$610$	22.6922	0.918780
$611$	7.84389	0.317330
$612$	3.93652	0.159124
$613$	−12.2708	−0.495613	−0.247807	−	0.968810i	$-0.579710\pi$
$613$	−12.2708	−0.495613	−0.247807	+	0.968810i	$0.579710\pi$
$614$	5.48727	0.221448
$615$	−18.0447	−0.727634
$616$	−2.17796	−0.0877524
$617$	−40.7754	−1.64155	−0.820777	−	0.571249i	$-0.806459\pi$
$617$	−40.7754	−1.64155	−0.820777	+	0.571249i	$0.806459\pi$
$618$	3.64130	0.146474
$619$	44.5182	1.78934	0.894668	−	0.446732i	$-0.147412\pi$
$619$	44.5182	1.78934	0.894668	+	0.446732i	$0.147412\pi$
$620$	−59.3030	−2.38167
$621$	6.14357	0.246533
$622$	−71.0629	−2.84936
$623$	36.0304	1.44353
$624$	−0.889944	−0.0356263
$625$	−14.4322	−0.577287
$626$	−35.4967	−1.41873
$627$	−1.61102	−0.0643378
$628$	−8.47596	−0.338227
$629$	12.0040	0.478630
$630$	−7.72100	−0.307612
$631$	−11.4019	−0.453904	−0.226952	−	0.973906i	$-0.572876\pi$
$631$	−11.4019	−0.453904	−0.226952	+	0.973906i	$0.572876\pi$
$632$	−25.3007	−1.00641
$633$	−41.9124	−1.66587
$634$	−2.47870	−0.0984417
$635$	−4.83405	−0.191833
$636$	−33.3436	−1.32216
$637$	−4.83164	−0.191436
$638$	4.26741	0.168948
$639$	1.24932	0.0494222
$640$	−35.6917	−1.41084
$641$	−33.8745	−1.33796	−0.668981	−	0.743279i	$-0.733269\pi$
$641$	−33.8745	−1.33796	−0.668981	+	0.743279i	$0.733269\pi$
$642$	20.1100	0.793677
$643$	6.76136	0.266642	0.133321	−	0.991073i	$-0.457436\pi$
$643$	6.76136	0.266642	0.133321	+	0.991073i	$0.457436\pi$
$644$	−12.7997	−0.504377
$645$	5.08046	0.200043
$646$	−26.2544	−1.03296
$647$	−24.7140	−0.971609	−0.485805	−	0.874067i	$-0.661473\pi$
$647$	−24.7140	−0.971609	−0.485805	+	0.874067i	$0.661473\pi$
$648$	22.4839	0.883251
$649$	−1.77398	−0.0696349
$650$	−3.70620	−0.145369
$651$	51.8075	2.03050
$652$	−49.5163	−1.93921
$653$	−43.0580	−1.68499	−0.842494	−	0.538706i	$-0.818913\pi$
$653$	−43.0580	−1.68499	−0.842494	+	0.538706i	$0.818913\pi$
$654$	11.1285	0.435159
$655$	23.9387	0.935364
$656$	3.52064	0.137458
$657$	3.07921	0.120131
$658$	62.4602	2.43495
$659$	35.3577	1.37734	0.688671	−	0.725074i	$-0.258195\pi$
$659$	35.3577	1.37734	0.688671	+	0.725074i	$0.258195\pi$
$660$	−1.96027	−0.0763035
$661$	−4.73902	−0.184326	−0.0921632	−	0.995744i	$-0.529378\pi$
$661$	−4.73902	−0.184326	−0.0921632	+	0.995744i	$0.529378\pi$
$662$	10.0240	0.389594
$663$	3.50478	0.136114
$664$	46.2254	1.79390
$665$	32.2774	1.25166
$666$	−6.55907	−0.254159
$667$	10.1475	0.392914
$668$	−11.6735	−0.451659
$669$	−43.6115	−1.68612
$670$	65.4605	2.52896
$671$	1.06990	0.0413031
$672$	26.9613	1.04005
$673$	35.5536	1.37049	0.685246	−	0.728312i	$-0.259695\pi$
$673$	35.5536	1.37049	0.685246	+	0.728312i	$0.259695\pi$
$674$	−42.1287	−1.62274
$675$	8.87888	0.341748
$676$	3.35919	0.129200
$677$	16.1619	0.621152	0.310576	−	0.950549i	$-0.399478\pi$
$677$	16.1619	0.621152	0.310576	+	0.950549i	$0.399478\pi$
$678$	−11.6282	−0.446580
$679$	34.6290	1.32894
$680$	−12.9260	−0.495689
$681$	30.0392	1.15110
$682$	−4.46077	−0.170812
$683$	36.5455	1.39837	0.699187	−	0.714938i	$-0.253545\pi$
$683$	36.5455	1.39837	0.699187	+	0.714938i	$0.253545\pi$
$684$	8.99197	0.343817
$685$	12.3986	0.473726
$686$	17.2665	0.659237
$687$	−14.3498	−0.547477
$688$	−0.991229	−0.0377903
$689$	6.31061	0.240415
$690$	−7.43663	−0.283108
$691$	16.5493	0.629564	0.314782	−	0.949164i	$-0.398069\pi$
$691$	16.5493	0.629564	0.314782	+	0.949164i	$0.398069\pi$
$692$	26.6743	1.01400
$693$	−0.364033	−0.0138285
$694$	−37.2592	−1.41434
$695$	34.9241	1.32475
$696$	45.3373	1.71851
$697$	−13.8650	−0.525175
$698$	−34.6558	−1.31174
$699$	34.7414	1.31404
$700$	−18.4985	−0.699177
$701$	7.59904	0.287012	0.143506	−	0.989649i	$-0.454162\pi$
$701$	7.59904	0.287012	0.143506	+	0.989649i	$0.454162\pi$
$702$	−12.8389	−0.484575
$703$	27.4200	1.03417
$704$	−2.54915	−0.0960748
$705$	22.7466	0.856687
$706$	−41.2553	−1.55266
$707$	3.31727	0.124759
$708$	−46.5795	−1.75056
$709$	−43.5734	−1.63643	−0.818217	−	0.574909i	$-0.805037\pi$
$709$	−43.5734	−1.63643	−0.818217	+	0.574909i	$0.805037\pi$
$710$	−10.1386	−0.380495
$711$	−4.22888	−0.158595
$712$	−32.9593	−1.23520
$713$	−10.6073	−0.397246
$714$	27.9083	1.04444
$715$	0.371001	0.0138746
$716$	38.8018	1.45009
$717$	32.9805	1.23168
$718$	−16.7259	−0.624204
$719$	15.0362	0.560754	0.280377	−	0.959890i	$-0.409540\pi$
$719$	15.0362	0.560754	0.280377	+	0.959890i	$0.409540\pi$
$720$	0.548602	0.0204452
$721$	−3.43971	−0.128102
$722$	−15.9866	−0.594958
$723$	18.3191	0.681293
$724$	−17.7316	−0.658989
$725$	14.6655	0.544664
$726$	39.9068	1.48108
$727$	4.03748	0.149742	0.0748710	−	0.997193i	$-0.476145\pi$
$727$	4.03748	0.149742	0.0748710	+	0.997193i	$0.476145\pi$
$728$	10.8231	0.401132
$729$	29.9283	1.10845
$730$	−24.9888	−0.924876
$731$	3.90366	0.144382
$732$	28.0925	1.03833
$733$	14.8206	0.547411	0.273706	−	0.961814i	$-0.411751\pi$
$733$	14.8206	0.547411	0.273706	+	0.961814i	$0.411751\pi$
$734$	55.6462	2.05394
$735$	−14.0113	−0.516816
$736$	−5.52018	−0.203476
$737$	3.08636	0.113688
$738$	7.57597	0.278875
$739$	17.0145	0.625890	0.312945	−	0.949771i	$-0.398684\pi$
$739$	17.0145	0.625890	0.312945	+	0.949771i	$0.398684\pi$
$740$	33.3644	1.22650
$741$	8.00578	0.294100
$742$	50.2509	1.84477
$743$	20.7225	0.760236	0.380118	−	0.924938i	$-0.375883\pi$
$743$	20.7225	0.760236	0.380118	+	0.924938i	$0.375883\pi$
$744$	−47.3915	−1.73746
$745$	27.6951	1.01467
$746$	−12.8582	−0.470774
$747$	7.72632	0.282691
$748$	−1.50621	−0.0550725
$749$	−18.9967	−0.694123
$750$	−44.3141	−1.61812
$751$	44.4871	1.62336	0.811679	−	0.584104i	$-0.198554\pi$
$751$	44.4871	1.62336	0.811679	+	0.584104i	$0.198554\pi$
$752$	−4.43801	−0.161837
$753$	−27.2035	−0.991350
$754$	−21.2065	−0.772294
$755$	−17.6114	−0.640944
$756$	−64.0821	−2.33064
$757$	−35.1996	−1.27935	−0.639675	−	0.768646i	$-0.720931\pi$
$757$	−35.1996	−1.27935	−0.639675	+	0.768646i	$0.720931\pi$
$758$	72.8160	2.64480
$759$	−0.350626	−0.0127269
$760$	−29.5261	−1.07103
$761$	0.158334	0.00573959	0.00286979	−	0.999996i	$-0.499087\pi$
$761$	0.158334	0.00573959	0.00286979	+	0.999996i	$0.499087\pi$
$762$	−9.54748	−0.345869
$763$	−10.5124	−0.380576
$764$	−54.6865	−1.97849
$765$	−2.16051	−0.0781133
$766$	−5.68027	−0.205236
$767$	8.81562	0.318314
$768$	−30.6422	−1.10570
$769$	33.2918	1.20053	0.600267	−	0.799800i	$-0.295061\pi$
$769$	33.2918	1.20053	0.600267	+	0.799800i	$0.295061\pi$
$770$	2.95425	0.106464
$771$	49.0584	1.76679
$772$	−61.2843	−2.20567
$773$	19.8921	0.715470	0.357735	−	0.933823i	$-0.383549\pi$
$773$	19.8921	0.715470	0.357735	+	0.933823i	$0.383549\pi$
$774$	−2.13300	−0.0766690
$775$	−15.3300	−0.550670
$776$	−31.6773	−1.13715
$777$	−29.1474	−1.04566
$778$	−23.1347	−0.829418
$779$	−31.6711	−1.13473
$780$	9.74137	0.348797
$781$	−0.478020	−0.0171049
$782$	−5.71407	−0.204335
$783$	50.8040	1.81559
$784$	2.73370	0.0976321
$785$	4.65192	0.166034
$786$	47.2802	1.68643
$787$	−23.0886	−0.823021	−0.411511	−	0.911405i	$-0.634999\pi$
$787$	−23.0886	−0.823021	−0.411511	+	0.911405i	$0.634999\pi$
$788$	41.5113	1.47878
$789$	−36.1270	−1.28616
$790$	34.3187	1.22100
$791$	10.9845	0.390564
$792$	0.333004	0.0118328
$793$	−5.31677	−0.188804
$794$	24.2803	0.861676
$795$	18.3002	0.649042
$796$	−56.1711	−1.99093
$797$	−34.6543	−1.22752	−0.613760	−	0.789493i	$-0.710344\pi$
$797$	−34.6543	−1.22752	−0.613760	+	0.789493i	$0.710344\pi$
$798$	63.7494	2.25670
$799$	17.4778	0.618319
$800$	−7.97794	−0.282063
$801$	−5.50896	−0.194649
$802$	−17.5689	−0.620379
$803$	−1.17818	−0.0415772
$804$	81.0387	2.85801
$805$	7.02493	0.247596
$806$	22.1673	0.780811
$807$	3.95083	0.139076
$808$	−3.03452	−0.106754
$809$	3.79330	0.133365	0.0666827	−	0.997774i	$-0.478758\pi$
$809$	3.79330	0.133365	0.0666827	+	0.997774i	$0.478758\pi$
$810$	−30.4978	−1.07158
$811$	11.6402	0.408742	0.204371	−	0.978894i	$-0.434485\pi$
$811$	11.6402	0.408742	0.204371	+	0.978894i	$0.434485\pi$
$812$	−105.846	−3.71448
$813$	45.9952	1.61312
$814$	2.50967	0.0879638
$815$	27.1764	0.951948
$816$	−1.98298	−0.0694180
$817$	8.91693	0.311964
$818$	78.3079	2.73797
$819$	1.80903	0.0632125
$820$	−38.5371	−1.34577
$821$	−23.2988	−0.813132	−0.406566	−	0.913621i	$-0.633274\pi$
$821$	−23.2988	−0.813132	−0.406566	+	0.913621i	$0.633274\pi$
$822$	24.4878	0.854110
$823$	−39.9730	−1.39337	−0.696686	−	0.717376i	$-0.745343\pi$
$823$	−39.9730	−1.39337	−0.696686	+	0.717376i	$0.745343\pi$
$824$	3.14652	0.109614
$825$	−0.506736	−0.0176423
$826$	70.1981	2.44250
$827$	42.4607	1.47650	0.738252	−	0.674525i	$-0.235652\pi$
$827$	42.4607	1.47650	0.738252	+	0.674525i	$0.235652\pi$
$828$	1.95704	0.0680117
$829$	−42.4835	−1.47551	−0.737757	−	0.675066i	$-0.764115\pi$
$829$	−42.4835	−1.47551	−0.737757	+	0.675066i	$0.764115\pi$
$830$	−62.7016	−2.17640
$831$	33.4115	1.15903
$832$	12.6678	0.439175
$833$	−10.7659	−0.373015
$834$	68.9767	2.38847
$835$	6.40683	0.221717
$836$	−3.44055	−0.118994
$837$	−53.1059	−1.83561
$838$	−34.0421	−1.17596
$839$	−42.3876	−1.46338	−0.731691	−	0.681636i	$-0.761269\pi$
$839$	−42.3876	−1.46338	−0.731691	+	0.681636i	$0.761269\pi$
$840$	31.3862	1.08293
$841$	54.9146	1.89361
$842$	−3.98300	−0.137263
$843$	−23.9456	−0.824730
$844$	−89.5099	−3.08106
$845$	−1.84365	−0.0634235
$846$	−9.55001	−0.328336
$847$	−37.6976	−1.29530
$848$	−3.57049	−0.122611
$849$	31.0934	1.06712
$850$	−8.25815	−0.283252
$851$	5.96776	0.204572
$852$	−12.5514	−0.430003
$853$	48.4560	1.65910	0.829550	−	0.558432i	$-0.188597\pi$
$853$	48.4560	1.65910	0.829550	+	0.558432i	$0.188597\pi$
$854$	−42.3370	−1.44874
$855$	−4.93513	−0.168778
$856$	17.3774	0.593949
$857$	−18.1626	−0.620421	−0.310211	−	0.950668i	$-0.600400\pi$
$857$	−18.1626	−0.620421	−0.310211	+	0.950668i	$0.600400\pi$
$858$	0.732744	0.0250155
$859$	−21.5387	−0.734891	−0.367445	−	0.930045i	$-0.619767\pi$
$859$	−21.5387	−0.734891	−0.367445	+	0.930045i	$0.619767\pi$
$860$	10.8500	0.369983
$861$	33.6662	1.14734
$862$	16.1913	0.551477
$863$	4.56049	0.155241	0.0776204	−	0.996983i	$-0.475268\pi$
$863$	4.56049	0.155241	0.0776204	+	0.996983i	$0.475268\pi$
$864$	−27.6370	−0.940231
$865$	−14.6399	−0.497770
$866$	26.7698	0.909675
$867$	−18.9303	−0.642906
$868$	110.642	3.75544
$869$	1.61807	0.0548894
$870$	−61.4969	−2.08494
$871$	−15.3374	−0.519687
$872$	9.61639	0.325652
$873$	−5.29468	−0.179198
$874$	−13.0523	−0.441502
$875$	41.8608	1.41515
$876$	−30.9356	−1.04522
$877$	−48.5675	−1.64001	−0.820004	−	0.572358i	$-0.806029\pi$
$877$	−48.5675	−1.64001	−0.820004	+	0.572358i	$0.806029\pi$
$878$	17.9508	0.605809
$879$	−14.4211	−0.486413
$880$	−0.209909	−0.00707603
$881$	15.7996	0.532302	0.266151	−	0.963931i	$-0.414248\pi$
$881$	15.7996	0.532302	0.266151	+	0.963931i	$0.414248\pi$
$882$	5.88257	0.198076
$883$	37.2690	1.25420	0.627101	−	0.778938i	$-0.284241\pi$
$883$	37.2690	1.25420	0.627101	+	0.778938i	$0.284241\pi$
$884$	7.48496	0.251746
$885$	25.5645	0.859343
$886$	−92.4556	−3.10611
$887$	28.0815	0.942884	0.471442	−	0.881897i	$-0.343734\pi$
$887$	28.0815	0.942884	0.471442	+	0.881897i	$0.343734\pi$
$888$	26.6629	0.894749
$889$	9.01893	0.302485
$890$	44.7069	1.49858
$891$	−1.43793	−0.0481724
$892$	−93.1384	−3.11851
$893$	39.9235	1.33599
$894$	54.6992	1.82941
$895$	−21.2959	−0.711843
$896$	66.5904	2.22463
$897$	1.74240	0.0581770
$898$	−67.8317	−2.26357
$899$	−87.7167	−2.92552
$900$	2.82837	0.0942790
$901$	14.0613	0.468451
$902$	−2.89876	−0.0965180
$903$	−9.47866	−0.315430
$904$	−10.0482	−0.334199
$905$	9.73175	0.323494
$906$	−34.7833	−1.15560
$907$	13.4535	0.446716	0.223358	−	0.974737i	$-0.428298\pi$
$907$	13.4535	0.446716	0.223358	+	0.974737i	$0.428298\pi$
$908$	64.1529	2.12899
$909$	−0.507202	−0.0168228
$910$	−14.6808	−0.486665
$911$	−48.1945	−1.59675	−0.798377	−	0.602158i	$-0.794308\pi$
$911$	−48.1945	−1.59675	−0.798377	+	0.602158i	$0.794308\pi$
$912$	−4.52960	−0.149990
$913$	−2.95629	−0.0978388
$914$	−68.0501	−2.25090
$915$	−15.4182	−0.509710
$916$	−30.6459	−1.01257
$917$	−44.6627	−1.47489
$918$	−28.6078	−0.944196
$919$	2.80512	0.0925325	0.0462663	−	0.998929i	$-0.485268\pi$
$919$	2.80512	0.0925325	0.0462663	+	0.998929i	$0.485268\pi$
$920$	−6.42615	−0.211864
$921$	−3.72832	−0.122852
$922$	5.33771	0.175788
$923$	2.37547	0.0781896
$924$	3.65730	0.120316
$925$	8.62480	0.283582
$926$	27.8994	0.916830
$927$	0.525924	0.0172736
$928$	−45.6489	−1.49850
$929$	−24.7523	−0.812098	−0.406049	−	0.913851i	$-0.633094\pi$
$929$	−24.7523	−0.812098	−0.406049	+	0.913851i	$0.633094\pi$
$930$	64.2833	2.10793
$931$	−24.5919	−0.805966
$932$	74.1953	2.43035
$933$	48.2836	1.58074
$934$	−26.6705	−0.872686
$935$	0.826665	0.0270348
$936$	−1.65483	−0.0540898
$937$	14.2970	0.467064	0.233532	−	0.972349i	$-0.424972\pi$
$937$	14.2970	0.467064	0.233532	+	0.972349i	$0.424972\pi$
$938$	−122.130	−3.98769
$939$	24.1182	0.787067
$940$	48.5786	1.58446
$941$	−53.4371	−1.74200	−0.871000	−	0.491284i	$-0.836528\pi$
$941$	−53.4371	−1.74200	−0.871000	+	0.491284i	$0.836528\pi$
$942$	9.18777	0.299354
$943$	−6.89298	−0.224466
$944$	−4.98780	−0.162339
$945$	35.1707	1.14410
$946$	0.816138	0.0265349
$947$	−41.6926	−1.35483	−0.677414	−	0.735602i	$-0.736899\pi$
$947$	−41.6926	−1.35483	−0.677414	+	0.735602i	$0.736899\pi$
$948$	42.4858	1.37987
$949$	5.85486	0.190057
$950$	−18.8636	−0.612018
$951$	1.68415	0.0546123
$952$	24.1161	0.781609
$953$	−1.74699	−0.0565907	−0.0282953	−	0.999600i	$-0.509008\pi$
$953$	−1.74699	−0.0565907	−0.0282953	+	0.999600i	$0.509008\pi$
$954$	−7.68323	−0.248754
$955$	30.0140	0.971231
$956$	70.4345	2.27801
$957$	−2.89949	−0.0937272
$958$	−33.5678	−1.08453
$959$	−23.1322	−0.746976
$960$	36.7354	1.18563
$961$	60.6911	1.95778
$962$	−12.4715	−0.402098
$963$	2.90454	0.0935975
$964$	39.1230	1.26007
$965$	33.6351	1.08275
$966$	13.8746	0.446407
$967$	37.3789	1.20202	0.601012	−	0.799240i	$-0.294764\pi$
$967$	37.3789	1.20202	0.601012	+	0.799240i	$0.294764\pi$
$968$	34.4843	1.10837
$969$	17.8385	0.573055
$970$	42.9680	1.37962
$971$	−6.00080	−0.192575	−0.0962874	−	0.995354i	$-0.530697\pi$
$971$	−6.00080	−0.192575	−0.0962874	+	0.995354i	$0.530697\pi$
$972$	18.1345	0.581665
$973$	−65.1582	−2.08887
$974$	−61.8777	−1.98269
$975$	2.51817	0.0806460
$976$	3.00818	0.0962896
$977$	50.0749	1.60204	0.801019	−	0.598639i	$-0.204291\pi$
$977$	50.0749	1.60204	0.801019	+	0.598639i	$0.204291\pi$
$978$	53.6747	1.71633
$979$	2.10787	0.0673677
$980$	−29.9232	−0.955861
$981$	1.60733	0.0513179
$982$	23.9200	0.763319
$983$	−43.9328	−1.40124	−0.700620	−	0.713535i	$-0.747093\pi$
$983$	−43.9328	−1.40124	−0.700620	+	0.713535i	$0.747093\pi$
$984$	−30.7966	−0.981760
$985$	−22.7829	−0.725925
$986$	−47.2523	−1.50482
$987$	−42.4386	−1.35083
$988$	17.0975	0.543944
$989$	1.94070	0.0617107
$990$	−0.451697	−0.0143559
$991$	−26.7911	−0.851048	−0.425524	−	0.904947i	$-0.639910\pi$
$991$	−26.7911	−0.851048	−0.425524	+	0.904947i	$0.639910\pi$
$992$	47.7173	1.51502
$993$	−6.81080	−0.216134
$994$	18.9157	0.599969
$995$	30.8288	0.977339
$996$	−77.6232	−2.45959
$997$	37.3999	1.18447	0.592233	−	0.805767i	$-0.298247\pi$
$997$	37.3999	1.18447	0.592233	+	0.805767i	$0.298247\pi$
$998$	46.4008	1.46879
$999$	29.8779	0.945294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.2	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.2	✓	21	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31499 q^{2} +1.57292 q^{3} +3.35919 q^{4} -1.84365 q^{5} -3.64130 q^{6} +3.43971 q^{7} -3.14652 q^{8} -0.525924 q^{9} +O(q^{10})\)	\(q-2.31499 q^{2} +1.57292 q^{3} +3.35919 q^{4} -1.84365 q^{5} -3.64130 q^{6} +3.43971 q^{7} -3.14652 q^{8} -0.525924 q^{9} +4.26804 q^{10} +0.201232 q^{11} +5.28374 q^{12} -1.00000 q^{13} -7.96292 q^{14} -2.89991 q^{15} +0.565791 q^{16} -2.22820 q^{17} +1.21751 q^{18} -5.08976 q^{19} -6.19318 q^{20} +5.41040 q^{21} -0.465850 q^{22} -1.10775 q^{23} -4.94923 q^{24} -1.60095 q^{25} +2.31499 q^{26} -5.54599 q^{27} +11.5547 q^{28} -9.16049 q^{29} +6.71328 q^{30} +9.57554 q^{31} +4.98324 q^{32} +0.316521 q^{33} +5.15827 q^{34} -6.34163 q^{35} -1.76668 q^{36} -5.38729 q^{37} +11.7828 q^{38} -1.57292 q^{39} +5.80109 q^{40} +6.22251 q^{41} -12.5250 q^{42} -1.75193 q^{43} +0.675976 q^{44} +0.969619 q^{45} +2.56443 q^{46} -7.84389 q^{47} +0.889944 q^{48} +4.83164 q^{49} +3.70620 q^{50} -3.50478 q^{51} -3.35919 q^{52} -6.31061 q^{53} +12.8389 q^{54} -0.371001 q^{55} -10.8231 q^{56} -8.00578 q^{57} +21.2065 q^{58} -8.81562 q^{59} -9.74137 q^{60} +5.31677 q^{61} -22.1673 q^{62} -1.80903 q^{63} -12.6678 q^{64} +1.84365 q^{65} -0.732744 q^{66} +15.3374 q^{67} -7.48496 q^{68} -1.74240 q^{69} +14.6808 q^{70} -2.37547 q^{71} +1.65483 q^{72} -5.85486 q^{73} +12.4715 q^{74} -2.51817 q^{75} -17.0975 q^{76} +0.692179 q^{77} +3.64130 q^{78} +8.04086 q^{79} -1.04312 q^{80} -7.14563 q^{81} -14.4051 q^{82} -14.6910 q^{83} +18.1746 q^{84} +4.10803 q^{85} +4.05572 q^{86} -14.4087 q^{87} -0.633180 q^{88} +10.4748 q^{89} -2.24466 q^{90} -3.43971 q^{91} -3.72114 q^{92} +15.0616 q^{93} +18.1586 q^{94} +9.38374 q^{95} +7.83824 q^{96} +10.0674 q^{97} -11.1852 q^{98} -0.105832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

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