LMFDB - Embedded newform 1339.2.a.e.1.8

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			1.8
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)

$f(q)$	$=$	$q-0.791299 q^{2} +0.744814 q^{3} -1.37385 q^{4} -0.897428 q^{5} -0.589370 q^{6} -1.66332 q^{7} +2.66972 q^{8} -2.44525 q^{9} +O(q^{10})$	\(q-0.791299 q^{2} +0.744814 q^{3} -1.37385 q^{4} -0.897428 q^{5} -0.589370 q^{6} -1.66332 q^{7} +2.66972 q^{8} -2.44525 q^{9} +0.710134 q^{10} +2.09766 q^{11} -1.02326 q^{12} -1.00000 q^{13} +1.31618 q^{14} -0.668416 q^{15} +0.635142 q^{16} +7.99798 q^{17} +1.93493 q^{18} +5.60749 q^{19} +1.23293 q^{20} -1.23886 q^{21} -1.65988 q^{22} +1.59871 q^{23} +1.98844 q^{24} -4.19462 q^{25} +0.791299 q^{26} -4.05570 q^{27} +2.28514 q^{28} -6.29240 q^{29} +0.528917 q^{30} -2.61576 q^{31} -5.84203 q^{32} +1.56237 q^{33} -6.32880 q^{34} +1.49271 q^{35} +3.35940 q^{36} -5.66394 q^{37} -4.43721 q^{38} -0.744814 q^{39} -2.39588 q^{40} -5.14989 q^{41} +0.980310 q^{42} -11.5950 q^{43} -2.88186 q^{44} +2.19444 q^{45} -1.26506 q^{46} -7.49666 q^{47} +0.473062 q^{48} -4.23337 q^{49} +3.31920 q^{50} +5.95701 q^{51} +1.37385 q^{52} +11.6663 q^{53} +3.20927 q^{54} -1.88250 q^{55} -4.44059 q^{56} +4.17654 q^{57} +4.97917 q^{58} -10.9389 q^{59} +0.918301 q^{60} +7.91638 q^{61} +2.06985 q^{62} +4.06723 q^{63} +3.35251 q^{64} +0.897428 q^{65} -1.23630 q^{66} -9.01249 q^{67} -10.9880 q^{68} +1.19074 q^{69} -1.18118 q^{70} +2.07732 q^{71} -6.52814 q^{72} -9.34909 q^{73} +4.48187 q^{74} -3.12421 q^{75} -7.70383 q^{76} -3.48908 q^{77} +0.589370 q^{78} -4.72678 q^{79} -0.569994 q^{80} +4.31502 q^{81} +4.07510 q^{82} +8.97215 q^{83} +1.70200 q^{84} -7.17761 q^{85} +9.17508 q^{86} -4.68666 q^{87} +5.60017 q^{88} +10.7502 q^{89} -1.73646 q^{90} +1.66332 q^{91} -2.19638 q^{92} -1.94826 q^{93} +5.93211 q^{94} -5.03232 q^{95} -4.35122 q^{96} +13.6217 q^{97} +3.34987 q^{98} -5.12931 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.791299	−0.559533	−0.279767	−	0.960068i	$-0.590257\pi$
$2$	−0.791299	−0.559533	−0.279767	+	0.960068i	$0.590257\pi$
$3$	0.744814	0.430018	0.215009	−	0.976612i	$-0.431022\pi$
$3$	0.744814	0.430018	0.215009	+	0.976612i	$0.431022\pi$
$4$	−1.37385	−0.686923
$5$	−0.897428	−0.401342	−0.200671	−	0.979659i	$-0.564312\pi$
$5$	−0.897428	−0.401342	−0.200671	+	0.979659i	$0.564312\pi$
$6$	−0.589370	−0.240609
$7$	−1.66332	−0.628675	−0.314337	−	0.949311i	$-0.601782\pi$
$7$	−1.66332	−0.628675	−0.314337	+	0.949311i	$0.601782\pi$
$8$	2.66972	0.943889
$9$	−2.44525	−0.815084
$10$	0.710134	0.224564
$11$	2.09766	0.632469	0.316234	−	0.948681i	$-0.397581\pi$
$11$	2.09766	0.632469	0.316234	+	0.948681i	$0.397581\pi$
$12$	−1.02326	−0.295389
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.31618	0.351764
$15$	−0.668416	−0.172584
$16$	0.635142	0.158785
$17$	7.99798	1.93980	0.969898	−	0.243512i	$-0.0782996\pi$
$17$	7.99798	1.93980	0.969898	+	0.243512i	$0.0782996\pi$
$18$	1.93493	0.456067
$19$	5.60749	1.28645	0.643224	−	0.765678i	$-0.277597\pi$
$19$	5.60749	1.28645	0.643224	+	0.765678i	$0.277597\pi$
$20$	1.23293	0.275691
$21$	−1.23886	−0.270342
$22$	−1.65988	−0.353887
$23$	1.59871	0.333355	0.166677	−	0.986011i	$-0.446696\pi$
$23$	1.59871	0.333355	0.166677	+	0.986011i	$0.446696\pi$
$24$	1.98844	0.405890
$25$	−4.19462	−0.838925
$26$	0.791299	0.155187
$27$	−4.05570	−0.780519
$28$	2.28514	0.431851
$29$	−6.29240	−1.16847	−0.584235	−	0.811585i	$-0.698605\pi$
$29$	−6.29240	−1.16847	−0.584235	+	0.811585i	$0.698605\pi$
$30$	0.528917	0.0965667
$31$	−2.61576	−0.469805	−0.234902	−	0.972019i	$-0.575477\pi$
$31$	−2.61576	−0.469805	−0.234902	+	0.972019i	$0.575477\pi$
$32$	−5.84203	−1.03273
$33$	1.56237	0.271973
$34$	−6.32880	−1.08538
$35$	1.49271	0.252314
$36$	3.35940	0.559900
$37$	−5.66394	−0.931147	−0.465573	−	0.885009i	$-0.654152\pi$
$37$	−5.66394	−0.931147	−0.465573	+	0.885009i	$0.654152\pi$
$38$	−4.43721	−0.719810
$39$	−0.744814	−0.119266
$40$	−2.39588	−0.378822
$41$	−5.14989	−0.804277	−0.402138	−	0.915579i	$-0.631733\pi$
$41$	−5.14989	−0.804277	−0.402138	+	0.915579i	$0.631733\pi$
$42$	0.980310	0.151265
$43$	−11.5950	−1.76821	−0.884107	−	0.467284i	$-0.845232\pi$
$43$	−11.5950	−1.76821	−0.884107	+	0.467284i	$0.845232\pi$
$44$	−2.88186	−0.434457
$45$	2.19444	0.327127
$46$	−1.26506	−0.186523
$47$	−7.49666	−1.09350	−0.546750	−	0.837296i	$-0.684135\pi$
$47$	−7.49666	−1.09350	−0.546750	+	0.837296i	$0.684135\pi$
$48$	0.473062	0.0682807
$49$	−4.23337	−0.604768
$50$	3.31920	0.469406
$51$	5.95701	0.834148
$52$	1.37385	0.190518
$53$	11.6663	1.60250	0.801248	−	0.598333i	$-0.204170\pi$
$53$	11.6663	1.60250	0.801248	+	0.598333i	$0.204170\pi$
$54$	3.20927	0.436726
$55$	−1.88250	−0.253836
$56$	−4.44059	−0.593399
$57$	4.17654	0.553196
$58$	4.97917	0.653797
$59$	−10.9389	−1.42412	−0.712060	−	0.702119i	$-0.752238\pi$
$59$	−10.9389	−1.42412	−0.712060	+	0.702119i	$0.752238\pi$
$60$	0.918301	0.118552
$61$	7.91638	1.01359	0.506794	−	0.862067i	$-0.330830\pi$
$61$	7.91638	1.01359	0.506794	+	0.862067i	$0.330830\pi$
$62$	2.06985	0.262871
$63$	4.06723	0.512423
$64$	3.35251	0.419064
$65$	0.897428	0.111312
$66$	−1.23630	−0.152178
$67$	−9.01249	−1.10105	−0.550526	−	0.834818i	$-0.685573\pi$
$67$	−9.01249	−1.10105	−0.550526	+	0.834818i	$0.685573\pi$
$68$	−10.9880	−1.33249
$69$	1.19074	0.143349
$70$	−1.18118	−0.141178
$71$	2.07732	0.246532	0.123266	−	0.992374i	$-0.460663\pi$
$71$	2.07732	0.246532	0.123266	+	0.992374i	$0.460663\pi$
$72$	−6.52814	−0.769349
$73$	−9.34909	−1.09423	−0.547114	−	0.837058i	$-0.684274\pi$
$73$	−9.34909	−1.09423	−0.547114	+	0.837058i	$0.684274\pi$
$74$	4.48187	0.521007
$75$	−3.12421	−0.360753
$76$	−7.70383	−0.883690
$77$	−3.48908	−0.397617
$78$	0.589370	0.0667331
$79$	−4.72678	−0.531805	−0.265902	−	0.964000i	$-0.585670\pi$
$79$	−4.72678	−0.531805	−0.265902	+	0.964000i	$0.585670\pi$
$80$	−0.569994	−0.0637273
$81$	4.31502	0.479447
$82$	4.07510	0.450020
$83$	8.97215	0.984821	0.492411	−	0.870363i	$-0.336116\pi$
$83$	8.97215	0.984821	0.492411	+	0.870363i	$0.336116\pi$
$84$	1.70200	0.185704
$85$	−7.17761	−0.778521
$86$	9.17508	0.989374
$87$	−4.68666	−0.502463
$88$	5.60017	0.596981
$89$	10.7502	1.13952	0.569758	−	0.821812i	$-0.307037\pi$
$89$	10.7502	1.13952	0.569758	+	0.821812i	$0.307037\pi$
$90$	−1.73646	−0.183039
$91$	1.66332	0.174363
$92$	−2.19638	−0.228989
$93$	−1.94826	−0.202025
$94$	5.93211	0.611850
$95$	−5.03232	−0.516305
$96$	−4.35122	−0.444095
$97$	13.6217	1.38307	0.691537	−	0.722341i	$-0.256934\pi$
$97$	13.6217	1.38307	0.691537	+	0.722341i	$0.256934\pi$
$98$	3.34987	0.338388
$99$	−5.12931	−0.515515
$100$	5.76276	0.576276
$101$	−17.8447	−1.77562	−0.887809	−	0.460212i	$-0.847773\pi$
$101$	−17.8447	−1.77562	−0.887809	+	0.460212i	$0.847773\pi$
$102$	−4.71377	−0.466733
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−2.66972	−0.261788
$105$	1.11179	0.108499
$106$	−9.23157	−0.896649
$107$	−9.49082	−0.917512	−0.458756	−	0.888562i	$-0.651705\pi$
$107$	−9.49082	−0.917512	−0.458756	+	0.888562i	$0.651705\pi$
$108$	5.57190	0.536157
$109$	−0.0473814	−0.00453831	−0.00226916	−	0.999997i	$-0.500722\pi$
$109$	−0.0473814	−0.00453831	−0.00226916	+	0.999997i	$0.500722\pi$
$110$	1.48962	0.142030
$111$	−4.21858	−0.400410
$112$	−1.05644	−0.0998245
$113$	−16.4681	−1.54919	−0.774594	−	0.632459i	$-0.782046\pi$
$113$	−16.4681	−1.54919	−0.774594	+	0.632459i	$0.782046\pi$
$114$	−3.30489	−0.309531
$115$	−1.43473	−0.133789
$116$	8.64478	0.802648
$117$	2.44525	0.226064
$118$	8.65592	0.796842
$119$	−13.3032	−1.21950
$120$	−1.78449	−0.162901
$121$	−6.59981	−0.599983
$122$	−6.26423	−0.567136
$123$	−3.83570	−0.345854
$124$	3.59365	0.322720
$125$	8.25151	0.738038
$126$	−3.21840	−0.286718
$127$	−7.77801	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$127$	−7.77801	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$128$	9.03122	0.798255
$129$	−8.63608	−0.760364
$130$	−0.710134	−0.0622829
$131$	12.1106	1.05810	0.529052	−	0.848589i	$-0.322548\pi$
$131$	12.1106	1.05810	0.529052	+	0.848589i	$0.322548\pi$
$132$	−2.14645	−0.186825
$133$	−9.32704	−0.808757
$134$	7.13158	0.616075
$135$	3.63970	0.313255
$136$	21.3524	1.83095
$137$	12.9138	1.10330	0.551650	−	0.834076i	$-0.313998\pi$
$137$	12.9138	1.10330	0.551650	+	0.834076i	$0.313998\pi$
$138$	−0.942234	−0.0802083
$139$	13.0722	1.10877	0.554383	−	0.832262i	$-0.312954\pi$
$139$	13.0722	1.10877	0.554383	+	0.832262i	$0.312954\pi$
$140$	−2.05075	−0.173320
$141$	−5.58362	−0.470225
$142$	−1.64378	−0.137943
$143$	−2.09766	−0.175415
$144$	−1.55308	−0.129424
$145$	5.64697	0.468956
$146$	7.39793	0.612257
$147$	−3.15308	−0.260061
$148$	7.78138	0.639626
$149$	−10.8955	−0.892591	−0.446296	−	0.894886i	$-0.647257\pi$
$149$	−10.8955	−0.892591	−0.446296	+	0.894886i	$0.647257\pi$
$150$	2.47219	0.201853
$151$	18.4952	1.50511	0.752557	−	0.658527i	$-0.228820\pi$
$151$	18.4952	1.50511	0.752557	+	0.658527i	$0.228820\pi$
$152$	14.9704	1.21426
$153$	−19.5571	−1.58110
$154$	2.76091	0.222480
$155$	2.34746	0.188552
$156$	1.02326	0.0819263
$157$	−7.62478	−0.608523	−0.304262	−	0.952588i	$-0.598410\pi$
$157$	−7.62478	−0.608523	−0.304262	+	0.952588i	$0.598410\pi$
$158$	3.74030	0.297562
$159$	8.68925	0.689103
$160$	5.24280	0.414480
$161$	−2.65917	−0.209572
$162$	−3.41447	−0.268266
$163$	15.3624	1.20327	0.601637	−	0.798770i	$-0.294515\pi$
$163$	15.3624	1.20327	0.601637	+	0.798770i	$0.294515\pi$
$164$	7.07515	0.552476
$165$	−1.40211	−0.109154
$166$	−7.09965	−0.551040
$167$	−14.6984	−1.13739	−0.568697	−	0.822547i	$-0.692552\pi$
$167$	−14.6984	−1.13739	−0.568697	+	0.822547i	$0.692552\pi$
$168$	−3.30742	−0.255173
$169$	1.00000	0.0769231
$170$	5.67964	0.435608
$171$	−13.7117	−1.04856
$172$	15.9297	1.21463
$173$	−1.40682	−0.106958	−0.0534791	−	0.998569i	$-0.517031\pi$
$173$	−1.40682	−0.106958	−0.0534791	+	0.998569i	$0.517031\pi$
$174$	3.70855	0.281145
$175$	6.97699	0.527411
$176$	1.33231	0.100427
$177$	−8.14742	−0.612398
$178$	−8.50661	−0.637597
$179$	−20.4662	−1.52972	−0.764860	−	0.644197i	$-0.777192\pi$
$179$	−20.4662	−1.52972	−0.764860	+	0.644197i	$0.777192\pi$
$180$	−3.01482	−0.224711
$181$	4.15103	0.308544	0.154272	−	0.988028i	$-0.450697\pi$
$181$	4.15103	0.308544	0.154272	+	0.988028i	$0.450697\pi$
$182$	−1.31618	−0.0975619
$183$	5.89623	0.435862
$184$	4.26812	0.314650
$185$	5.08298	0.373708
$186$	1.54165	0.113040
$187$	16.7771	1.22686
$188$	10.2993	0.751151
$189$	6.74591	0.490693
$190$	3.98207	0.288890
$191$	−7.82163	−0.565953	−0.282977	−	0.959127i	$-0.591322\pi$
$191$	−7.82163	−0.565953	−0.282977	+	0.959127i	$0.591322\pi$
$192$	2.49700	0.180205
$193$	−4.48624	−0.322927	−0.161463	−	0.986879i	$-0.551621\pi$
$193$	−4.48624	−0.322927	−0.161463	+	0.986879i	$0.551621\pi$
$194$	−10.7788	−0.773875
$195$	0.668416	0.0478663
$196$	5.81600	0.415429
$197$	−17.7890	−1.26741	−0.633707	−	0.773573i	$-0.718467\pi$
$197$	−17.7890	−1.26741	−0.633707	+	0.773573i	$0.718467\pi$
$198$	4.05882	0.288448
$199$	−27.3197	−1.93664	−0.968321	−	0.249709i	$-0.919665\pi$
$199$	−27.3197	−1.93664	−0.968321	+	0.249709i	$0.919665\pi$
$200$	−11.1985	−0.791852
$201$	−6.71263	−0.473472
$202$	14.1205	0.993517
$203$	10.4663	0.734587
$204$	−8.18401	−0.572995
$205$	4.62165	0.322790
$206$	0.791299	0.0551324
$207$	−3.90926	−0.271712
$208$	−0.635142	−0.0440392
$209$	11.7626	0.813638
$210$	−0.879758	−0.0607090
$211$	−5.44345	−0.374742	−0.187371	−	0.982289i	$-0.559997\pi$
$211$	−5.44345	−0.374742	−0.187371	+	0.982289i	$0.559997\pi$
$212$	−16.0278	−1.10079
$213$	1.54722	0.106013
$214$	7.51008	0.513378
$215$	10.4056	0.709658
$216$	−10.8276	−0.736724
$217$	4.35084	0.295355
$218$	0.0374928	0.00253934
$219$	−6.96333	−0.470538
$220$	2.58626	0.174366
$221$	−7.99798	−0.538003
$222$	3.33816	0.224043
$223$	−0.429907	−0.0287887	−0.0143943	−	0.999896i	$-0.504582\pi$
$223$	−0.429907	−0.0287887	−0.0143943	+	0.999896i	$0.504582\pi$
$224$	9.71715	0.649254
$225$	10.2569	0.683794
$226$	13.0312	0.866822
$227$	−5.02280	−0.333375	−0.166687	−	0.986010i	$-0.553307\pi$
$227$	−5.02280	−0.333375	−0.166687	+	0.986010i	$0.553307\pi$
$228$	−5.73792	−0.380003
$229$	−7.79356	−0.515013	−0.257507	−	0.966277i	$-0.582901\pi$
$229$	−7.79356	−0.515013	−0.257507	+	0.966277i	$0.582901\pi$
$230$	1.13530	0.0748595
$231$	−2.59871	−0.170983
$232$	−16.7990	−1.10291
$233$	−26.7331	−1.75134	−0.875671	−	0.482908i	$-0.839581\pi$
$233$	−26.7331	−1.75134	−0.875671	+	0.482908i	$0.839581\pi$
$234$	−1.93493	−0.126490
$235$	6.72772	0.438868
$236$	15.0283	0.978260
$237$	−3.52057	−0.228686
$238$	10.5268	0.682351
$239$	1.78258	0.115305	0.0576526	−	0.998337i	$-0.481638\pi$
$239$	1.78258	0.115305	0.0576526	+	0.998337i	$0.481638\pi$
$240$	−0.424539	−0.0274039
$241$	18.4303	1.18720	0.593599	−	0.804761i	$-0.297706\pi$
$241$	18.4303	1.18720	0.593599	+	0.804761i	$0.297706\pi$
$242$	5.22243	0.335710
$243$	15.3810	0.986690
$244$	−10.8759	−0.696257
$245$	3.79915	0.242719
$246$	3.03519	0.193517
$247$	−5.60749	−0.356796
$248$	−6.98336	−0.443444
$249$	6.68258	0.423491
$250$	−6.52941	−0.412956
$251$	4.04114	0.255074	0.127537	−	0.991834i	$-0.459293\pi$
$251$	4.04114	0.255074	0.127537	+	0.991834i	$0.459293\pi$
$252$	−5.58775	−0.351995
$253$	3.35356	0.210836
$254$	6.15473	0.386182
$255$	−5.34598	−0.334778
$256$	−13.8514	−0.865714
$257$	9.51842	0.593743	0.296871	−	0.954917i	$-0.404057\pi$
$257$	9.51842	0.593743	0.296871	+	0.954917i	$0.404057\pi$
$258$	6.83372	0.425449
$259$	9.42094	0.585389
$260$	−1.23293	−0.0764629
$261$	15.3865	0.952401
$262$	−9.58308	−0.592045
$263$	−1.55336	−0.0957844	−0.0478922	−	0.998853i	$-0.515250\pi$
$263$	−1.55336	−0.0957844	−0.0478922	+	0.998853i	$0.515250\pi$
$264$	4.17109	0.256713
$265$	−10.4697	−0.643149
$266$	7.38048	0.452526
$267$	8.00688	0.490013
$268$	12.3818	0.756337
$269$	−3.06432	−0.186835	−0.0934175	−	0.995627i	$-0.529779\pi$
$269$	−3.06432	−0.186835	−0.0934175	+	0.995627i	$0.529779\pi$
$270$	−2.88009	−0.175277
$271$	0.624876	0.0379585	0.0189793	−	0.999820i	$-0.493958\pi$
$271$	0.624876	0.0379585	0.0189793	+	0.999820i	$0.493958\pi$
$272$	5.07985	0.308011
$273$	1.23886	0.0749793
$274$	−10.2187	−0.617333
$275$	−8.79890	−0.530594
$276$	−1.63590	−0.0984694
$277$	−28.7016	−1.72451	−0.862256	−	0.506472i	$-0.830949\pi$
$277$	−28.7016	−1.72451	−0.862256	+	0.506472i	$0.830949\pi$
$278$	−10.3440	−0.620391
$279$	6.39620	0.382931
$280$	3.98511	0.238156
$281$	−26.8693	−1.60289	−0.801445	−	0.598068i	$-0.795935\pi$
$281$	−26.8693	−1.60289	−0.801445	+	0.598068i	$0.795935\pi$
$282$	4.41831	0.263107
$283$	12.5454	0.745745	0.372873	−	0.927883i	$-0.378373\pi$
$283$	12.5454	0.745745	0.372873	+	0.927883i	$0.378373\pi$
$284$	−2.85392	−0.169349
$285$	−3.74814	−0.222021
$286$	1.65988	0.0981507
$287$	8.56589	0.505629
$288$	14.2852	0.841766
$289$	46.9677	2.76281
$290$	−4.46845	−0.262396
$291$	10.1456	0.594747
$292$	12.8442	0.751650
$293$	8.66612	0.506280	0.253140	−	0.967430i	$-0.418537\pi$
$293$	8.66612	0.506280	0.253140	+	0.967430i	$0.418537\pi$
$294$	2.49503	0.145513
$295$	9.81685	0.571559
$296$	−15.1212	−0.878899
$297$	−8.50748	−0.493654
$298$	8.62157	0.499434
$299$	−1.59871	−0.0924559
$300$	4.29218	0.247809
$301$	19.2861	1.11163
$302$	−14.6352	−0.842162
$303$	−13.2910	−0.763548
$304$	3.56155	0.204269
$305$	−7.10438	−0.406796
$306$	15.4755	0.884676
$307$	22.5899	1.28927	0.644636	−	0.764489i	$-0.277009\pi$
$307$	22.5899	1.28927	0.644636	+	0.764489i	$0.277009\pi$
$308$	4.79345	0.273132
$309$	−0.744814	−0.0423710
$310$	−1.85754	−0.105501
$311$	14.1412	0.801876	0.400938	−	0.916105i	$-0.368684\pi$
$311$	14.1412	0.801876	0.400938	+	0.916105i	$0.368684\pi$
$312$	−1.98844	−0.112574
$313$	20.9686	1.18522	0.592608	−	0.805491i	$-0.298098\pi$
$313$	20.9686	1.18522	0.592608	+	0.805491i	$0.298098\pi$
$314$	6.03348	0.340489
$315$	−3.65005	−0.205657
$316$	6.49387	0.365309
$317$	−11.8709	−0.666734	−0.333367	−	0.942797i	$-0.608185\pi$
$317$	−11.8709	−0.666734	−0.333367	+	0.942797i	$0.608185\pi$
$318$	−6.87580	−0.385576
$319$	−13.1993	−0.739020
$320$	−3.00864	−0.168188
$321$	−7.06889	−0.394547
$322$	2.10420	0.117262
$323$	44.8486	2.49544
$324$	−5.92817	−0.329343
$325$	4.19462	0.232676
$326$	−12.1562	−0.673272
$327$	−0.0352903	−0.00195156
$328$	−13.7488	−0.759148
$329$	12.4693	0.687457
$330$	1.10949	0.0610754
$331$	2.34863	0.129092	0.0645461	−	0.997915i	$-0.479440\pi$
$331$	2.34863	0.129092	0.0645461	+	0.997915i	$0.479440\pi$
$332$	−12.3263	−0.676496
$333$	13.8498	0.758963
$334$	11.6308	0.636410
$335$	8.08806	0.441898
$336$	−0.786853	−0.0429263
$337$	0.527374	0.0287279	0.0143639	−	0.999897i	$-0.495428\pi$
$337$	0.527374	0.0287279	0.0143639	+	0.999897i	$0.495428\pi$
$338$	−0.791299	−0.0430410
$339$	−12.2657	−0.666179
$340$	9.86093	0.534784
$341$	−5.48699	−0.297137
$342$	10.8501	0.586706
$343$	18.6847	1.00888
$344$	−30.9553	−1.66900
$345$	−1.06861	−0.0575318
$346$	1.11321	0.0598467
$347$	−3.20145	−0.171863	−0.0859313	−	0.996301i	$-0.527387\pi$
$347$	−3.20145	−0.171863	−0.0859313	+	0.996301i	$0.527387\pi$
$348$	6.43875	0.345153
$349$	16.0738	0.860410	0.430205	−	0.902731i	$-0.358441\pi$
$349$	16.0738	0.860410	0.430205	+	0.902731i	$0.358441\pi$
$350$	−5.52089	−0.295104
$351$	4.05570	0.216477
$352$	−12.2546	−0.653173
$353$	31.7673	1.69080	0.845401	−	0.534132i	$-0.179361\pi$
$353$	31.7673	1.69080	0.845401	+	0.534132i	$0.179361\pi$
$354$	6.44705	0.342657
$355$	−1.86424	−0.0989438
$356$	−14.7691	−0.782760
$357$	−9.90839	−0.524408
$358$	16.1949	0.855928
$359$	−25.4135	−1.34127	−0.670636	−	0.741786i	$-0.733979\pi$
$359$	−25.4135	−1.34127	−0.670636	+	0.741786i	$0.733979\pi$
$360$	5.85854	0.308772
$361$	12.4440	0.654946
$362$	−3.28471	−0.172640
$363$	−4.91563	−0.258004
$364$	−2.28514	−0.119774
$365$	8.39013	0.439160
$366$	−4.66568	−0.243879
$367$	−17.9564	−0.937319	−0.468659	−	0.883379i	$-0.655263\pi$
$367$	−17.9564	−0.937319	−0.468659	+	0.883379i	$0.655263\pi$
$368$	1.01541	0.0529319
$369$	12.5928	0.655553
$370$	−4.02216	−0.209102
$371$	−19.4048	−1.00745
$372$	2.67660	0.138775
$373$	−27.3737	−1.41736	−0.708678	−	0.705532i	$-0.750708\pi$
$373$	−27.3737	−1.41736	−0.708678	+	0.705532i	$0.750708\pi$
$374$	−13.2757	−0.686469
$375$	6.14584	0.317370
$376$	−20.0140	−1.03214
$377$	6.29240	0.324075
$378$	−5.33804	−0.274559
$379$	−0.710007	−0.0364706	−0.0182353	−	0.999834i	$-0.505805\pi$
$379$	−0.710007	−0.0364706	−0.0182353	+	0.999834i	$0.505805\pi$
$380$	6.91363	0.354662
$381$	−5.79316	−0.296793
$382$	6.18925	0.316670
$383$	−9.97278	−0.509585	−0.254793	−	0.966996i	$-0.582007\pi$
$383$	−9.97278	−0.509585	−0.254793	+	0.966996i	$0.582007\pi$
$384$	6.72658	0.343264
$385$	3.13120	0.159581
$386$	3.54996	0.180688
$387$	28.3526	1.44124
$388$	−18.7141	−0.950064
$389$	−29.8517	−1.51354	−0.756772	−	0.653679i	$-0.773225\pi$
$389$	−29.8517	−1.51354	−0.756772	+	0.653679i	$0.773225\pi$
$390$	−0.528917	−0.0267828
$391$	12.7865	0.646640
$392$	−11.3019	−0.570834
$393$	9.02011	0.455004
$394$	14.0764	0.709160
$395$	4.24195	0.213436
$396$	7.04688	0.354119
$397$	−24.8483	−1.24710	−0.623550	−	0.781784i	$-0.714310\pi$
$397$	−24.8483	−1.24710	−0.623550	+	0.781784i	$0.714310\pi$
$398$	21.6181	1.08362
$399$	−6.94691	−0.347780
$400$	−2.66418	−0.133209
$401$	11.3456	0.566574	0.283287	−	0.959035i	$-0.408575\pi$
$401$	11.3456	0.566574	0.283287	+	0.959035i	$0.408575\pi$
$402$	5.31170	0.264923
$403$	2.61576	0.130300
$404$	24.5159	1.21971
$405$	−3.87242	−0.192422
$406$	−8.28194	−0.411026
$407$	−11.8810	−0.588921
$408$	15.9035	0.787343
$409$	18.8826	0.933685	0.466843	−	0.884340i	$-0.345391\pi$
$409$	18.8826	0.933685	0.466843	+	0.884340i	$0.345391\pi$
$410$	−3.65711	−0.180612
$411$	9.61837	0.474439
$412$	1.37385	0.0676845
$413$	18.1948	0.895309
$414$	3.09339	0.152032
$415$	−8.05185	−0.395250
$416$	5.84203	0.286429
$417$	9.73632	0.476790
$418$	−9.30776	−0.455257
$419$	9.85091	0.481248	0.240624	−	0.970618i	$-0.422648\pi$
$419$	9.85091	0.481248	0.240624	+	0.970618i	$0.422648\pi$
$420$	−1.52743	−0.0745308
$421$	21.3674	1.04139	0.520693	−	0.853744i	$-0.325674\pi$
$421$	21.3674	1.04139	0.520693	+	0.853744i	$0.325674\pi$
$422$	4.30740	0.209681
$423$	18.3312	0.891295
$424$	31.1459	1.51258
$425$	−33.5485	−1.62734
$426$	−1.22431	−0.0593181
$427$	−13.1675	−0.637218
$428$	13.0389	0.630260
$429$	−1.56237	−0.0754318
$430$	−8.23397	−0.397077
$431$	−39.5101	−1.90314	−0.951568	−	0.307438i	$-0.900528\pi$
$431$	−39.5101	−1.90314	−0.951568	+	0.307438i	$0.900528\pi$
$432$	−2.57594	−0.123935
$433$	4.07114	0.195647	0.0978233	−	0.995204i	$-0.468812\pi$
$433$	4.07114	0.195647	0.0978233	+	0.995204i	$0.468812\pi$
$434$	−3.44282	−0.165261
$435$	4.20594	0.201660
$436$	0.0650947	0.00311747
$437$	8.96477	0.428843
$438$	5.51008	0.263282
$439$	−24.9401	−1.19032	−0.595162	−	0.803606i	$-0.702912\pi$
$439$	−24.9401	−1.19032	−0.595162	+	0.803606i	$0.702912\pi$
$440$	−5.02575	−0.239593
$441$	10.3517	0.492937
$442$	6.32880	0.301030
$443$	28.3657	1.34770	0.673848	−	0.738870i	$-0.264640\pi$
$443$	28.3657	1.34770	0.673848	+	0.738870i	$0.264640\pi$
$444$	5.79568	0.275051
$445$	−9.64751	−0.457336
$446$	0.340185	0.0161082
$447$	−8.11509	−0.383831
$448$	−5.57629	−0.263455
$449$	10.8751	0.513227	0.256613	−	0.966514i	$-0.417393\pi$
$449$	10.8751	0.513227	0.256613	+	0.966514i	$0.417393\pi$
$450$	−8.11629	−0.382606
$451$	−10.8027	−0.508680
$452$	22.6246	1.06417
$453$	13.7754	0.647227
$454$	3.97453	0.186534
$455$	−1.49271	−0.0699792
$456$	11.1502	0.522155
$457$	−12.9345	−0.605049	−0.302524	−	0.953142i	$-0.597829\pi$
$457$	−12.9345	−0.605049	−0.302524	+	0.953142i	$0.597829\pi$
$458$	6.16704	0.288167
$459$	−32.4374	−1.51405
$460$	1.97110	0.0919028
$461$	40.2893	1.87646	0.938230	−	0.346011i	$-0.112464\pi$
$461$	40.2893	1.87646	0.938230	+	0.346011i	$0.112464\pi$
$462$	2.05636	0.0956705
$463$	16.6491	0.773748	0.386874	−	0.922133i	$-0.373555\pi$
$463$	16.6491	0.773748	0.386874	+	0.922133i	$0.373555\pi$
$464$	−3.99657	−0.185536
$465$	1.74842	0.0810810
$466$	21.1539	0.979934
$467$	9.23792	0.427480	0.213740	−	0.976891i	$-0.431435\pi$
$467$	9.23792	0.427480	0.213740	+	0.976891i	$0.431435\pi$
$468$	−3.35940	−0.155288
$469$	14.9906	0.692203
$470$	−5.32364	−0.245561
$471$	−5.67904	−0.261676
$472$	−29.2037	−1.34421
$473$	−24.3223	−1.11834
$474$	2.78583	0.127957
$475$	−23.5213	−1.07923
$476$	18.2765	0.837703
$477$	−28.5272	−1.30617
$478$	−1.41055	−0.0645171
$479$	−16.5837	−0.757730	−0.378865	−	0.925452i	$-0.623686\pi$
$479$	−16.5837	−0.757730	−0.378865	+	0.925452i	$0.623686\pi$
$480$	3.90491	0.178234
$481$	5.66394	0.258254
$482$	−14.5839	−0.664277
$483$	−1.98058	−0.0901197
$484$	9.06712	0.412142
$485$	−12.2245	−0.555085
$486$	−12.1710	−0.552086
$487$	35.1015	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$487$	35.1015	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$488$	21.1345	0.956715
$489$	11.4421	0.517430
$490$	−3.00626	−0.135809
$491$	35.4893	1.60161	0.800805	−	0.598925i	$-0.204405\pi$
$491$	35.4893	1.60161	0.800805	+	0.598925i	$0.204405\pi$
$492$	5.26967	0.237575
$493$	−50.3265	−2.26659
$494$	4.43721	0.199639
$495$	4.60319	0.206898
$496$	−1.66138	−0.0745982
$497$	−3.45524	−0.154989
$498$	−5.28792	−0.236957
$499$	20.4938	0.917428	0.458714	−	0.888584i	$-0.348310\pi$
$499$	20.4938	0.917428	0.458714	+	0.888584i	$0.348310\pi$
$500$	−11.3363	−0.506975
$501$	−10.9475	−0.489100
$502$	−3.19775	−0.142723
$503$	30.4805	1.35906	0.679528	−	0.733649i	$-0.262184\pi$
$503$	30.4805	1.35906	0.679528	+	0.733649i	$0.262184\pi$
$504$	10.8584	0.483670
$505$	16.0144	0.712630
$506$	−2.65367	−0.117970
$507$	0.744814	0.0330783
$508$	10.6858	0.474105
$509$	−12.0209	−0.532815	−0.266407	−	0.963860i	$-0.585837\pi$
$509$	−12.0209	−0.532815	−0.266407	+	0.963860i	$0.585837\pi$
$510$	4.23027	0.187320
$511$	15.5505	0.687914
$512$	−7.10182	−0.313859
$513$	−22.7423	−1.00410
$514$	−7.53192	−0.332219
$515$	0.897428	0.0395454
$516$	11.8646	0.522312
$517$	−15.7255	−0.691605
$518$	−7.45478	−0.327544
$519$	−1.04782	−0.0459940
$520$	2.39588	0.105066
$521$	−32.2674	−1.41366	−0.706830	−	0.707383i	$-0.749875\pi$
$521$	−32.2674	−1.41366	−0.706830	+	0.707383i	$0.749875\pi$
$522$	−12.1753	−0.532900
$523$	9.06250	0.396275	0.198138	−	0.980174i	$-0.436511\pi$
$523$	9.06250	0.396275	0.198138	+	0.980174i	$0.436511\pi$
$524$	−16.6380	−0.726836
$525$	5.19656	0.226796
$526$	1.22917	0.0535945
$527$	−20.9208	−0.911325
$528$	0.992325	0.0431854
$529$	−20.4441	−0.888875
$530$	8.28467	0.359863
$531$	26.7483	1.16078
$532$	12.8139	0.555554
$533$	5.14989	0.223066
$534$	−6.33584	−0.274179
$535$	8.51732	0.368236
$536$	−24.0608	−1.03927
$537$	−15.2435	−0.657807
$538$	2.42480	0.104540
$539$	−8.88019	−0.382497
$540$	−5.00038	−0.215182
$541$	24.0439	1.03373	0.516863	−	0.856068i	$-0.327100\pi$
$541$	24.0439	1.03373	0.516863	+	0.856068i	$0.327100\pi$
$542$	−0.494464	−0.0212391
$543$	3.09174	0.132679
$544$	−46.7245	−2.00329
$545$	0.0425214	0.00182141
$546$	−0.980310	−0.0419534
$547$	3.49189	0.149302	0.0746512	−	0.997210i	$-0.476216\pi$
$547$	3.49189	0.149302	0.0746512	+	0.997210i	$0.476216\pi$
$548$	−17.7416	−0.757882
$549$	−19.3575	−0.826160
$550$	6.96257	0.296885
$551$	−35.2846	−1.50317
$552$	3.17895	0.135305
$553$	7.86214	0.334332
$554$	22.7116	0.964922
$555$	3.78587	0.160701
$556$	−17.9591	−0.761636
$557$	−0.693937	−0.0294031	−0.0147015	−	0.999892i	$-0.504680\pi$
$557$	−0.693937	−0.0294031	−0.0147015	+	0.999892i	$0.504680\pi$
$558$	−5.06131	−0.214262
$559$	11.5950	0.490414
$560$	0.948081	0.0400637
$561$	12.4958	0.527573
$562$	21.2617	0.896870
$563$	−46.4634	−1.95820	−0.979099	−	0.203385i	$-0.934806\pi$
$563$	−46.4634	−1.95820	−0.979099	+	0.203385i	$0.934806\pi$
$564$	7.67103	0.323009
$565$	14.7789	0.621754
$566$	−9.92715	−0.417269
$567$	−7.17725	−0.301416
$568$	5.54586	0.232699
$569$	−10.9357	−0.458446	−0.229223	−	0.973374i	$-0.573619\pi$
$569$	−10.9357	−0.458446	−0.229223	+	0.973374i	$0.573619\pi$
$570$	2.96590	0.124228
$571$	18.1370	0.759012	0.379506	−	0.925189i	$-0.376094\pi$
$571$	18.1370	0.759012	0.379506	+	0.925189i	$0.376094\pi$
$572$	2.88186	0.120497
$573$	−5.82566	−0.243370
$574$	−6.77819	−0.282916
$575$	−6.70600	−0.279659
$576$	−8.19774	−0.341572
$577$	−13.0803	−0.544540	−0.272270	−	0.962221i	$-0.587774\pi$
$577$	−13.0803	−0.544540	−0.272270	+	0.962221i	$0.587774\pi$
$578$	−37.1655	−1.54588
$579$	−3.34141	−0.138864
$580$	−7.75807	−0.322136
$581$	−14.9235	−0.619132
$582$	−8.02822	−0.332781
$583$	24.4721	1.01353
$584$	−24.9595	−1.03283
$585$	−2.19444	−0.0907288
$586$	−6.85749	−0.283280
$587$	−6.49601	−0.268119	−0.134059	−	0.990973i	$-0.542801\pi$
$587$	−6.49601	−0.268119	−0.134059	+	0.990973i	$0.542801\pi$
$588$	4.33184	0.178642
$589$	−14.6679	−0.604379
$590$	−7.76807	−0.319806
$591$	−13.2495	−0.545011
$592$	−3.59741	−0.147853
$593$	−20.0791	−0.824551	−0.412276	−	0.911059i	$-0.635266\pi$
$593$	−20.0791	−0.824551	−0.412276	+	0.911059i	$0.635266\pi$
$594$	6.73197	0.276216
$595$	11.9386	0.489437
$596$	14.9687	0.613141
$597$	−20.3481	−0.832791
$598$	1.26506	0.0517322
$599$	20.0814	0.820503	0.410251	−	0.911972i	$-0.365441\pi$
$599$	20.0814	0.820503	0.410251	+	0.911972i	$0.365441\pi$
$600$	−8.34078	−0.340511
$601$	−23.1472	−0.944196	−0.472098	−	0.881546i	$-0.656503\pi$
$601$	−23.1472	−0.944196	−0.472098	+	0.881546i	$0.656503\pi$
$602$	−15.2611	−0.621995
$603$	22.0378	0.897450
$604$	−25.4095	−1.03390
$605$	5.92286	0.240798
$606$	10.5172	0.427230
$607$	1.97969	0.0803531	0.0401765	−	0.999193i	$-0.487208\pi$
$607$	1.97969	0.0803531	0.0401765	+	0.999193i	$0.487208\pi$
$608$	−32.7591	−1.32856
$609$	7.79541	0.315886
$610$	5.62169	0.227616
$611$	7.49666	0.303283
$612$	26.8684	1.08609
$613$	26.8275	1.08355	0.541775	−	0.840523i	$-0.317752\pi$
$613$	26.8275	1.08355	0.541775	+	0.840523i	$0.317752\pi$
$614$	−17.8754	−0.721391
$615$	3.44227	0.138806
$616$	−9.31487	−0.375307
$617$	19.7804	0.796327	0.398163	−	0.917314i	$-0.369648\pi$
$617$	19.7804	0.796327	0.398163	+	0.917314i	$0.369648\pi$
$618$	0.589370	0.0237080
$619$	−20.7074	−0.832299	−0.416150	−	0.909296i	$-0.636621\pi$
$619$	−20.7074	−0.832299	−0.416150	+	0.909296i	$0.636621\pi$
$620$	−3.22504	−0.129521
$621$	−6.48390	−0.260190
$622$	−11.1900	−0.448676
$623$	−17.8810	−0.716386
$624$	−0.473062	−0.0189377
$625$	13.5680	0.542719
$626$	−16.5925	−0.663168
$627$	8.76096	0.349879
$628$	10.4753	0.418008
$629$	−45.3001	−1.80623
$630$	2.88828	0.115072
$631$	36.8608	1.46741	0.733703	−	0.679470i	$-0.237790\pi$
$631$	36.8608	1.46741	0.733703	+	0.679470i	$0.237790\pi$
$632$	−12.6192	−0.501965
$633$	−4.05435	−0.161146
$634$	9.39340	0.373060
$635$	6.98020	0.277001
$636$	−11.9377	−0.473360
$637$	4.23337	0.167732
$638$	10.4446	0.413506
$639$	−5.07957	−0.200945
$640$	−8.10487	−0.320373
$641$	−10.0160	−0.395608	−0.197804	−	0.980242i	$-0.563381\pi$
$641$	−10.0160	−0.395608	−0.197804	+	0.980242i	$0.563381\pi$
$642$	5.59361	0.220762
$643$	43.2029	1.70376	0.851878	−	0.523741i	$-0.175464\pi$
$643$	43.2029	1.70376	0.851878	+	0.523741i	$0.175464\pi$
$644$	3.65328	0.143960
$645$	7.75026	0.305166
$646$	−35.4887	−1.39628
$647$	45.1966	1.77686	0.888431	−	0.459010i	$-0.151796\pi$
$647$	45.1966	1.77686	0.888431	+	0.459010i	$0.151796\pi$
$648$	11.5199	0.452544
$649$	−22.9461	−0.900712
$650$	−3.31920	−0.130190
$651$	3.24057	0.127008
$652$	−21.1055	−0.826556
$653$	8.34561	0.326589	0.163294	−	0.986577i	$-0.447788\pi$
$653$	8.34561	0.326589	0.163294	+	0.986577i	$0.447788\pi$
$654$	0.0279252	0.00109196
$655$	−10.8684	−0.424662
$656$	−3.27091	−0.127708
$657$	22.8609	0.891888
$658$	−9.86697	−0.384655
$659$	−6.14293	−0.239295	−0.119647	−	0.992816i	$-0.538176\pi$
$659$	−6.14293	−0.239295	−0.119647	+	0.992816i	$0.538176\pi$
$660$	1.92628	0.0749805
$661$	−31.2948	−1.21723	−0.608613	−	0.793467i	$-0.708274\pi$
$661$	−31.2948	−1.21723	−0.608613	+	0.793467i	$0.708274\pi$
$662$	−1.85847	−0.0722314
$663$	−5.95701	−0.231351
$664$	23.9531	0.929562
$665$	8.37035	0.324588
$666$	−10.9593	−0.424665
$667$	−10.0597	−0.389515
$668$	20.1933	0.781302
$669$	−0.320200	−0.0123797
$670$	−6.40008	−0.247257
$671$	16.6059	0.641063
$672$	7.23747	0.279191
$673$	−6.65219	−0.256423	−0.128212	−	0.991747i	$-0.540924\pi$
$673$	−6.65219	−0.256423	−0.128212	+	0.991747i	$0.540924\pi$
$674$	−0.417311	−0.0160742
$675$	17.0121	0.654797
$676$	−1.37385	−0.0528402
$677$	−12.8141	−0.492484	−0.246242	−	0.969208i	$-0.579196\pi$
$677$	−12.8141	−0.492484	−0.246242	+	0.969208i	$0.579196\pi$
$678$	9.70581	0.372749
$679$	−22.6572	−0.869503
$680$	−19.1622	−0.734838
$681$	−3.74105	−0.143357
$682$	4.34185	0.166258
$683$	−13.3734	−0.511721	−0.255860	−	0.966714i	$-0.582359\pi$
$683$	−13.3734	−0.511721	−0.255860	+	0.966714i	$0.582359\pi$
$684$	18.8378	0.720282
$685$	−11.5892	−0.442801
$686$	−14.7852	−0.564500
$687$	−5.80475	−0.221465
$688$	−7.36444	−0.280767
$689$	−11.6663	−0.444452
$690$	0.845587	0.0321910
$691$	7.93925	0.302023	0.151012	−	0.988532i	$-0.451747\pi$
$691$	7.93925	0.302023	0.151012	+	0.988532i	$0.451747\pi$
$692$	1.93275	0.0734720
$693$	8.53168	0.324092
$694$	2.53330	0.0961628
$695$	−11.7313	−0.444994
$696$	−12.5121	−0.474270
$697$	−41.1887	−1.56013
$698$	−12.7192	−0.481428
$699$	−19.9112	−0.753109
$700$	−9.58531	−0.362291
$701$	−15.0037	−0.566683	−0.283341	−	0.959019i	$-0.591443\pi$
$701$	−15.0037	−0.566683	−0.283341	+	0.959019i	$0.591443\pi$
$702$	−3.20927	−0.121126
$703$	−31.7605	−1.19787
$704$	7.03243	0.265045
$705$	5.01089	0.188721
$706$	−25.1374	−0.946060
$707$	29.6815	1.11629
$708$	11.1933	0.420670
$709$	−13.6861	−0.513993	−0.256997	−	0.966412i	$-0.582733\pi$
$709$	−13.6861	−0.513993	−0.256997	+	0.966412i	$0.582733\pi$
$710$	1.47518	0.0553623
$711$	11.5582	0.433466
$712$	28.7000	1.07558
$713$	−4.18185	−0.156612
$714$	7.84050	0.293423
$715$	1.88250	0.0704015
$716$	28.1175	1.05080
$717$	1.32769	0.0495834
$718$	20.1097	0.750487
$719$	2.65475	0.0990056	0.0495028	−	0.998774i	$-0.484236\pi$
$719$	2.65475	0.0990056	0.0495028	+	0.998774i	$0.484236\pi$
$720$	1.39378	0.0519431
$721$	1.66332	0.0619452
$722$	−9.84691	−0.366464
$723$	13.7271	0.510517
$724$	−5.70288	−0.211946
$725$	26.3942	0.980258
$726$	3.88974	0.144362
$727$	−13.0436	−0.483760	−0.241880	−	0.970306i	$-0.577764\pi$
$727$	−13.0436	−0.483760	−0.241880	+	0.970306i	$0.577764\pi$
$728$	4.44059	0.164579
$729$	−1.48909	−0.0551517
$730$	−6.63910	−0.245724
$731$	−92.7362	−3.42997
$732$	−8.10050	−0.299403
$733$	45.0063	1.66235	0.831173	−	0.556014i	$-0.187670\pi$
$733$	45.0063	1.66235	0.831173	+	0.556014i	$0.187670\pi$
$734$	14.2089	0.524461
$735$	2.82966	0.104373
$736$	−9.33973	−0.344267
$737$	−18.9052	−0.696381
$738$	−9.96465	−0.366804
$739$	−20.0984	−0.739331	−0.369666	−	0.929165i	$-0.620528\pi$
$739$	−20.0984	−0.739331	−0.369666	+	0.929165i	$0.620528\pi$
$740$	−6.98323	−0.256709
$741$	−4.17654	−0.153429
$742$	15.3550	0.563701
$743$	11.1186	0.407904	0.203952	−	0.978981i	$-0.434621\pi$
$743$	11.1186	0.407904	0.203952	+	0.978981i	$0.434621\pi$
$744$	−5.20130	−0.190689
$745$	9.77789	0.358234
$746$	21.6608	0.793057
$747$	−21.9392	−0.802712
$748$	−23.0491	−0.842758
$749$	15.7862	0.576817
$750$	−4.86320	−0.177579
$751$	−10.6554	−0.388822	−0.194411	−	0.980920i	$-0.562279\pi$
$751$	−10.6554	−0.388822	−0.194411	+	0.980920i	$0.562279\pi$
$752$	−4.76145	−0.173632
$753$	3.00989	0.109687
$754$	−4.97917	−0.181331
$755$	−16.5981	−0.604066
$756$	−9.26784	−0.337068
$757$	−48.3166	−1.75610	−0.878049	−	0.478571i	$-0.841155\pi$
$757$	−48.3166	−1.75610	−0.878049	+	0.478571i	$0.841155\pi$
$758$	0.561828	0.0204065
$759$	2.49778	0.0906635
$760$	−13.4349	−0.487335
$761$	−32.7060	−1.18559	−0.592796	−	0.805352i	$-0.701976\pi$
$761$	−32.7060	−1.18559	−0.592796	+	0.805352i	$0.701976\pi$
$762$	4.58413	0.166065
$763$	0.0788102	0.00285312
$764$	10.7457	0.388766
$765$	17.5511	0.634560
$766$	7.89145	0.285130
$767$	10.9389	0.394980
$768$	−10.3167	−0.372273
$769$	27.2288	0.981895	0.490947	−	0.871189i	$-0.336651\pi$
$769$	27.2288	0.981895	0.490947	+	0.871189i	$0.336651\pi$
$770$	−2.47771	−0.0892906
$771$	7.08945	0.255320
$772$	6.16340	0.221826
$773$	29.0560	1.04507	0.522535	−	0.852618i	$-0.324986\pi$
$773$	29.0560	1.04507	0.522535	+	0.852618i	$0.324986\pi$
$774$	−22.4354	−0.806423
$775$	10.9721	0.394131
$776$	36.3661	1.30547
$777$	7.01684	0.251728
$778$	23.6217	0.846878
$779$	−28.8779	−1.03466
$780$	−0.918301	−0.0328804
$781$	4.35751	0.155924
$782$	−10.1179	−0.361816
$783$	25.5201	0.912013
$784$	−2.68879	−0.0960284
$785$	6.84269	0.244226
$786$	−7.13761	−0.254590
$787$	38.4695	1.37129	0.685645	−	0.727937i	$-0.259520\pi$
$787$	38.4695	1.37129	0.685645	+	0.727937i	$0.259520\pi$
$788$	24.4393	0.870615
$789$	−1.15697	−0.0411891
$790$	−3.35665	−0.119424
$791$	27.3917	0.973935
$792$	−13.6938	−0.486589
$793$	−7.91638	−0.281119
$794$	19.6624	0.697794
$795$	−7.79798	−0.276566
$796$	37.5330	1.33032
$797$	−12.1800	−0.431438	−0.215719	−	0.976455i	$-0.569210\pi$
$797$	−12.1800	−0.431438	−0.215719	+	0.976455i	$0.569210\pi$
$798$	5.49708	0.194595
$799$	−59.9582	−2.12117
$800$	24.5051	0.866387
$801$	−26.2869	−0.928802
$802$	−8.97780	−0.317017
$803$	−19.6112	−0.692065
$804$	9.22211	0.325239
$805$	2.38641	0.0841099
$806$	−2.06985	−0.0729074
$807$	−2.28235	−0.0803425
$808$	−47.6405	−1.67599
$809$	−46.5004	−1.63487	−0.817433	−	0.576024i	$-0.804604\pi$
$809$	−46.5004	−1.63487	−0.817433	+	0.576024i	$0.804604\pi$
$810$	3.06424	0.107666
$811$	−3.93531	−0.138187	−0.0690937	−	0.997610i	$-0.522011\pi$
$811$	−3.93531	−0.138187	−0.0690937	+	0.997610i	$0.522011\pi$
$812$	−14.3790	−0.504605
$813$	0.465416	0.0163229
$814$	9.40146	0.329521
$815$	−13.7866	−0.482924
$816$	3.78354	0.132451
$817$	−65.0186	−2.27471
$818$	−14.9418	−0.522428
$819$	−4.06723	−0.142121
$820$	−6.34943	−0.221732
$821$	43.0729	1.50325	0.751627	−	0.659588i	$-0.229269\pi$
$821$	43.0729	1.50325	0.751627	+	0.659588i	$0.229269\pi$
$822$	−7.61101	−0.265465
$823$	−8.82086	−0.307476	−0.153738	−	0.988112i	$-0.549131\pi$
$823$	−8.82086	−0.307476	−0.153738	+	0.988112i	$0.549131\pi$
$824$	−2.66972	−0.0930042
$825$	−6.55354	−0.228165
$826$	−14.3975	−0.500955
$827$	3.21170	0.111682	0.0558409	−	0.998440i	$-0.482216\pi$
$827$	3.21170	0.111682	0.0558409	+	0.998440i	$0.482216\pi$
$828$	5.37071	0.186645
$829$	−25.1141	−0.872249	−0.436125	−	0.899886i	$-0.643649\pi$
$829$	−25.1141	−0.872249	−0.436125	+	0.899886i	$0.643649\pi$
$830$	6.37143	0.221155
$831$	−21.3773	−0.741572
$832$	−3.35251	−0.116227
$833$	−33.8585	−1.17313
$834$	−7.70434	−0.266780
$835$	13.1907	0.456484
$836$	−16.1600	−0.558906
$837$	10.6087	0.366692
$838$	−7.79502	−0.269274
$839$	40.2956	1.39116	0.695580	−	0.718449i	$-0.255147\pi$
$839$	40.2956	1.39116	0.695580	+	0.718449i	$0.255147\pi$
$840$	2.96817	0.102411
$841$	10.5943	0.365320
$842$	−16.9080	−0.582690
$843$	−20.0126	−0.689272
$844$	7.47846	0.257419
$845$	−0.897428	−0.0308725
$846$	−14.5055	−0.498709
$847$	10.9776	0.377194
$848$	7.40979	0.254453
$849$	9.34397	0.320684
$850$	26.5469	0.910552
$851$	−9.05502	−0.310402
$852$	−2.12564	−0.0728231
$853$	−38.0426	−1.30255	−0.651277	−	0.758840i	$-0.725767\pi$
$853$	−38.0426	−1.30255	−0.651277	+	0.758840i	$0.725767\pi$
$854$	10.4194	0.356544
$855$	12.3053	0.420832
$856$	−25.3378	−0.866030
$857$	15.5935	0.532664	0.266332	−	0.963881i	$-0.414188\pi$
$857$	15.5935	0.532664	0.266332	+	0.963881i	$0.414188\pi$
$858$	1.23630	0.0422066
$859$	50.6044	1.72660	0.863300	−	0.504691i	$-0.168394\pi$
$859$	50.6044	1.72660	0.863300	+	0.504691i	$0.168394\pi$
$860$	−14.2957	−0.487480
$861$	6.37999	0.217430
$862$	31.2643	1.06487
$863$	−8.90220	−0.303034	−0.151517	−	0.988455i	$-0.548416\pi$
$863$	−8.90220	−0.303034	−0.151517	+	0.988455i	$0.548416\pi$
$864$	23.6935	0.806070
$865$	1.26252	0.0429268
$866$	−3.22149	−0.109471
$867$	34.9822	1.18806
$868$	−5.97739	−0.202886
$869$	−9.91519	−0.336350
$870$	−3.32816	−0.112835
$871$	9.01249	0.305377
$872$	−0.126495	−0.00428366
$873$	−33.3085	−1.12732
$874$	−7.09382	−0.239952
$875$	−13.7249	−0.463986
$876$	9.56653	0.323223
$877$	17.6294	0.595303	0.297652	−	0.954675i	$-0.403797\pi$
$877$	17.6294	0.595303	0.297652	+	0.954675i	$0.403797\pi$
$878$	19.7351	0.666026
$879$	6.45464	0.217710
$880$	−1.19566	−0.0403055
$881$	−15.0317	−0.506430	−0.253215	−	0.967410i	$-0.581488\pi$
$881$	−15.0317	−0.506430	−0.253215	+	0.967410i	$0.581488\pi$
$882$	−8.19127	−0.275814
$883$	−9.88548	−0.332673	−0.166336	−	0.986069i	$-0.553194\pi$
$883$	−9.88548	−0.332673	−0.166336	+	0.986069i	$0.553194\pi$
$884$	10.9880	0.369566
$885$	7.31172	0.245781
$886$	−22.4458	−0.754080
$887$	56.5293	1.89807	0.949034	−	0.315174i	$-0.102063\pi$
$887$	56.5293	1.89807	0.949034	+	0.315174i	$0.102063\pi$
$888$	−11.2624	−0.377943
$889$	12.9373	0.433903
$890$	7.63407	0.255895
$891$	9.05145	0.303235
$892$	0.590626	0.0197756
$893$	−42.0375	−1.40673
$894$	6.42146	0.214766
$895$	18.3670	0.613940
$896$	−15.0218	−0.501843
$897$	−1.19074	−0.0397578
$898$	−8.60545	−0.287167
$899$	16.4594	0.548953
$900$	−14.0914	−0.469714
$901$	93.3072	3.10851
$902$	8.54818	0.284623
$903$	14.3645	0.478022
$904$	−43.9652	−1.46226
$905$	−3.72525	−0.123832
$906$	−10.9005	−0.362145
$907$	38.7866	1.28789	0.643943	−	0.765073i	$-0.277297\pi$
$907$	38.7866	1.28789	0.643943	+	0.765073i	$0.277297\pi$
$908$	6.90054	0.229003
$909$	43.6349	1.44728
$910$	1.18118	0.0391557
$911$	−18.9860	−0.629035	−0.314517	−	0.949252i	$-0.601843\pi$
$911$	−18.9860	−0.629035	−0.314517	+	0.949252i	$0.601843\pi$
$912$	2.65269	0.0878395
$913$	18.8205	0.622869
$914$	10.2350	0.338545
$915$	−5.29144	−0.174930
$916$	10.7072	0.353774
$917$	−20.1437	−0.665204
$918$	25.6677	0.847160
$919$	12.7929	0.422000	0.211000	−	0.977486i	$-0.432328\pi$
$919$	12.7929	0.422000	0.211000	+	0.977486i	$0.432328\pi$
$920$	−3.83033	−0.126282
$921$	16.8253	0.554411
$922$	−31.8809	−1.04994
$923$	−2.07732	−0.0683758
$924$	3.57023	0.117452
$925$	23.7581	0.781162
$926$	−13.1744	−0.432938
$927$	2.44525	0.0803126
$928$	36.7604	1.20672
$929$	−29.7727	−0.976812	−0.488406	−	0.872617i	$-0.662421\pi$
$929$	−29.7727	−0.976812	−0.488406	+	0.872617i	$0.662421\pi$
$930$	−1.38352	−0.0453675
$931$	−23.7386	−0.778002
$932$	36.7271	1.20304
$933$	10.5326	0.344821
$934$	−7.30996	−0.239189
$935$	−15.0562	−0.492391
$936$	6.52814	0.213379
$937$	49.9339	1.63127	0.815634	−	0.578568i	$-0.196388\pi$
$937$	49.9339	1.63127	0.815634	+	0.578568i	$0.196388\pi$
$938$	−11.8621	−0.387311
$939$	15.6177	0.509665
$940$	−9.24284	−0.301468
$941$	18.6329	0.607414	0.303707	−	0.952766i	$-0.401776\pi$
$941$	18.6329	0.607414	0.303707	+	0.952766i	$0.401776\pi$
$942$	4.49382	0.146416
$943$	−8.23319	−0.268109
$944$	−6.94774	−0.226130
$945$	−6.05397	−0.196936
$946$	19.2462	0.625748
$947$	−56.1505	−1.82465	−0.912324	−	0.409469i	$-0.865714\pi$
$947$	−56.1505	−1.82465	−0.912324	+	0.409469i	$0.865714\pi$
$948$	4.83672	0.157089
$949$	9.34909	0.303484
$950$	18.6124	0.603866
$951$	−8.84157	−0.286708
$952$	−35.5158	−1.15107
$953$	36.2933	1.17566	0.587828	−	0.808986i	$-0.299983\pi$
$953$	36.2933	1.17566	0.587828	+	0.808986i	$0.299983\pi$
$954$	22.5735	0.730845
$955$	7.01935	0.227141
$956$	−2.44898	−0.0792058
$957$	−9.83104	−0.317792
$958$	13.1227	0.423975
$959$	−21.4797	−0.693617
$960$	−2.24087	−0.0723239
$961$	−24.1578	−0.779283
$962$	−4.48187	−0.144501
$963$	23.2074	0.747850
$964$	−25.3204	−0.815514
$965$	4.02608	0.129604
$966$	1.56723	0.0504249
$967$	−14.8892	−0.478806	−0.239403	−	0.970920i	$-0.576952\pi$
$967$	−14.8892	−0.478806	−0.239403	+	0.970920i	$0.576952\pi$
$968$	−17.6197	−0.566317
$969$	33.4039	1.07309
$970$	9.67323	0.310589
$971$	47.8823	1.53662	0.768308	−	0.640080i	$-0.221099\pi$
$971$	47.8823	1.53662	0.768308	+	0.640080i	$0.221099\pi$
$972$	−21.1311	−0.677780
$973$	−21.7431	−0.697053
$974$	−27.7758	−0.889994
$975$	3.12421	0.100055
$976$	5.02803	0.160943
$977$	−48.9782	−1.56695	−0.783476	−	0.621423i	$-0.786555\pi$
$977$	−48.9782	−1.56695	−0.783476	+	0.621423i	$0.786555\pi$
$978$	−9.05413	−0.289519
$979$	22.5502	0.720709
$980$	−5.21944	−0.166729
$981$	0.115859	0.00369911
$982$	−28.0827	−0.896154
$983$	6.32121	0.201615	0.100808	−	0.994906i	$-0.467857\pi$
$983$	6.32121	0.201615	0.100808	+	0.994906i	$0.467857\pi$
$984$	−10.2403	−0.326448
$985$	15.9643	0.508666
$986$	39.8233	1.26823
$987$	9.28733	0.295619
$988$	7.70383	0.245091
$989$	−18.5370	−0.589442
$990$	−3.64250	−0.115766
$991$	−18.6791	−0.593360	−0.296680	−	0.954977i	$-0.595879\pi$
$991$	−18.6791	−0.593360	−0.296680	+	0.954977i	$0.595879\pi$
$992$	15.2814	0.485184
$993$	1.74929	0.0555120
$994$	2.73413	0.0867214
$995$	24.5175	0.777255
$996$	−9.18083	−0.290906
$997$	59.7155	1.89121	0.945605	−	0.325317i	$-0.105471\pi$
$997$	59.7155	1.89121	0.945605	+	0.325317i	$0.105471\pi$
$998$	−16.2167	−0.513331
$999$	22.9712	0.726778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.8	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-0.791299 q^{2} +0.744814 q^{3} -1.37385 q^{4} -0.897428 q^{5} -0.589370 q^{6} -1.66332 q^{7} +2.66972 q^{8} -2.44525 q^{9} +O(q^{10})\)	\(q-0.791299 q^{2} +0.744814 q^{3} -1.37385 q^{4} -0.897428 q^{5} -0.589370 q^{6} -1.66332 q^{7} +2.66972 q^{8} -2.44525 q^{9} +0.710134 q^{10} +2.09766 q^{11} -1.02326 q^{12} -1.00000 q^{13} +1.31618 q^{14} -0.668416 q^{15} +0.635142 q^{16} +7.99798 q^{17} +1.93493 q^{18} +5.60749 q^{19} +1.23293 q^{20} -1.23886 q^{21} -1.65988 q^{22} +1.59871 q^{23} +1.98844 q^{24} -4.19462 q^{25} +0.791299 q^{26} -4.05570 q^{27} +2.28514 q^{28} -6.29240 q^{29} +0.528917 q^{30} -2.61576 q^{31} -5.84203 q^{32} +1.56237 q^{33} -6.32880 q^{34} +1.49271 q^{35} +3.35940 q^{36} -5.66394 q^{37} -4.43721 q^{38} -0.744814 q^{39} -2.39588 q^{40} -5.14989 q^{41} +0.980310 q^{42} -11.5950 q^{43} -2.88186 q^{44} +2.19444 q^{45} -1.26506 q^{46} -7.49666 q^{47} +0.473062 q^{48} -4.23337 q^{49} +3.31920 q^{50} +5.95701 q^{51} +1.37385 q^{52} +11.6663 q^{53} +3.20927 q^{54} -1.88250 q^{55} -4.44059 q^{56} +4.17654 q^{57} +4.97917 q^{58} -10.9389 q^{59} +0.918301 q^{60} +7.91638 q^{61} +2.06985 q^{62} +4.06723 q^{63} +3.35251 q^{64} +0.897428 q^{65} -1.23630 q^{66} -9.01249 q^{67} -10.9880 q^{68} +1.19074 q^{69} -1.18118 q^{70} +2.07732 q^{71} -6.52814 q^{72} -9.34909 q^{73} +4.48187 q^{74} -3.12421 q^{75} -7.70383 q^{76} -3.48908 q^{77} +0.589370 q^{78} -4.72678 q^{79} -0.569994 q^{80} +4.31502 q^{81} +4.07510 q^{82} +8.97215 q^{83} +1.70200 q^{84} -7.17761 q^{85} +9.17508 q^{86} -4.68666 q^{87} +5.60017 q^{88} +10.7502 q^{89} -1.73646 q^{90} +1.66332 q^{91} -2.19638 q^{92} -1.94826 q^{93} +5.93211 q^{94} -5.03232 q^{95} -4.35122 q^{96} +13.6217 q^{97} +3.34987 q^{98} -5.12931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 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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