LMFDB - Embedded newform 1441.2.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1441,2,Mod(1,1441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1441, 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("1441.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1441 = 11 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1441.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:	$11.5064429313$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	1441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.262885 q^{2} +0.849379 q^{3} -1.93089 q^{4} -1.83441 q^{5} +0.223289 q^{6} -0.312976 q^{7} -1.03337 q^{8} -2.27856 q^{9} +O(q^{10})$	\(q+0.262885 q^{2} +0.849379 q^{3} -1.93089 q^{4} -1.83441 q^{5} +0.223289 q^{6} -0.312976 q^{7} -1.03337 q^{8} -2.27856 q^{9} -0.482239 q^{10} -1.00000 q^{11} -1.64006 q^{12} -1.17709 q^{13} -0.0822766 q^{14} -1.55811 q^{15} +3.59013 q^{16} +2.54628 q^{17} -0.598997 q^{18} +7.32694 q^{19} +3.54205 q^{20} -0.265835 q^{21} -0.262885 q^{22} -0.146051 q^{23} -0.877723 q^{24} -1.63493 q^{25} -0.309440 q^{26} -4.48349 q^{27} +0.604323 q^{28} +4.58112 q^{29} -0.409603 q^{30} +0.837277 q^{31} +3.01053 q^{32} -0.849379 q^{33} +0.669379 q^{34} +0.574127 q^{35} +4.39964 q^{36} +4.83508 q^{37} +1.92614 q^{38} -0.999799 q^{39} +1.89563 q^{40} +10.1987 q^{41} -0.0698840 q^{42} +8.06168 q^{43} +1.93089 q^{44} +4.17981 q^{45} -0.0383946 q^{46} -2.40165 q^{47} +3.04938 q^{48} -6.90205 q^{49} -0.429799 q^{50} +2.16276 q^{51} +2.27284 q^{52} +4.93783 q^{53} -1.17864 q^{54} +1.83441 q^{55} +0.323420 q^{56} +6.22335 q^{57} +1.20431 q^{58} +12.2163 q^{59} +3.00854 q^{60} -7.24820 q^{61} +0.220107 q^{62} +0.713133 q^{63} -6.38883 q^{64} +2.15928 q^{65} -0.223289 q^{66} -11.4402 q^{67} -4.91660 q^{68} -0.124053 q^{69} +0.150929 q^{70} -3.06764 q^{71} +2.35459 q^{72} +3.48730 q^{73} +1.27107 q^{74} -1.38868 q^{75} -14.1475 q^{76} +0.312976 q^{77} -0.262832 q^{78} +10.4102 q^{79} -6.58577 q^{80} +3.02749 q^{81} +2.68108 q^{82} -4.53584 q^{83} +0.513299 q^{84} -4.67093 q^{85} +2.11929 q^{86} +3.89110 q^{87} +1.03337 q^{88} +9.16636 q^{89} +1.09881 q^{90} +0.368402 q^{91} +0.282009 q^{92} +0.711165 q^{93} -0.631356 q^{94} -13.4406 q^{95} +2.55708 q^{96} -10.4679 q^{97} -1.81444 q^{98} +2.27856 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9}+O(q^{10}) $ 31 * q + 6 * q^2 + 4 * q^3 + 38 * q^4 + 8 * q^5 + 7 * q^6 + 4 * q^7 + 24 * q^8 + 45 * q^9	\( 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9} - 8 q^{10} - 31 q^{11} + 10 q^{12} - 8 q^{13} + 29 q^{14} + 36 q^{15} + 52 q^{16} - q^{17} + 33 q^{18} - 2 q^{19} + 22 q^{20} - 13 q^{21} - 6 q^{22} + 45 q^{23} + 16 q^{24} + 41 q^{25} + 24 q^{26} + 22 q^{27} + 17 q^{28} + 5 q^{29} + 29 q^{30} + 28 q^{31} + 69 q^{32} - 4 q^{33} + 14 q^{34} + 36 q^{35} + 63 q^{36} - 3 q^{37} + 4 q^{38} + 40 q^{39} - 48 q^{40} + 21 q^{41} - 9 q^{42} - 20 q^{43} - 38 q^{44} + 28 q^{45} - 24 q^{46} + 57 q^{47} - 46 q^{48} + 37 q^{49} + 64 q^{50} + 17 q^{51} - 11 q^{52} + 32 q^{53} - 26 q^{54} - 8 q^{55} + 84 q^{56} + 10 q^{57} - 17 q^{58} + 70 q^{59} - 33 q^{60} - 51 q^{61} - 34 q^{62} + 32 q^{63} + 80 q^{64} - q^{65} - 7 q^{66} + 24 q^{67} - 13 q^{68} + 19 q^{69} - 9 q^{70} + 128 q^{71} + 118 q^{72} - 27 q^{73} - 23 q^{74} + 41 q^{75} - 34 q^{76} - 4 q^{77} + 9 q^{78} + 2 q^{79} - 45 q^{80} + 43 q^{81} - 18 q^{82} + 46 q^{83} - 103 q^{84} - 50 q^{85} + 78 q^{86} - 9 q^{87} - 24 q^{88} + 52 q^{89} - 46 q^{90} + 38 q^{91} + 54 q^{92} + 4 q^{93} + 3 q^{94} + 70 q^{95} - 21 q^{96} + 3 q^{97} - 120 q^{98} - 45 q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 + 4 * q^3 + 38 * q^4 + 8 * q^5 + 7 * q^6 + 4 * q^7 + 24 * q^8 + 45 * q^9 - 8 * q^10 - 31 * q^11 + 10 * q^12 - 8 * q^13 + 29 * q^14 + 36 * q^15 + 52 * q^16 - q^17 + 33 * q^18 - 2 * q^19 + 22 * q^20 - 13 * q^21 - 6 * q^22 + 45 * q^23 + 16 * q^24 + 41 * q^25 + 24 * q^26 + 22 * q^27 + 17 * q^28 + 5 * q^29 + 29 * q^30 + 28 * q^31 + 69 * q^32 - 4 * q^33 + 14 * q^34 + 36 * q^35 + 63 * q^36 - 3 * q^37 + 4 * q^38 + 40 * q^39 - 48 * q^40 + 21 * q^41 - 9 * q^42 - 20 * q^43 - 38 * q^44 + 28 * q^45 - 24 * q^46 + 57 * q^47 - 46 * q^48 + 37 * q^49 + 64 * q^50 + 17 * q^51 - 11 * q^52 + 32 * q^53 - 26 * q^54 - 8 * q^55 + 84 * q^56 + 10 * q^57 - 17 * q^58 + 70 * q^59 - 33 * q^60 - 51 * q^61 - 34 * q^62 + 32 * q^63 + 80 * q^64 - q^65 - 7 * q^66 + 24 * q^67 - 13 * q^68 + 19 * q^69 - 9 * q^70 + 128 * q^71 + 118 * q^72 - 27 * q^73 - 23 * q^74 + 41 * q^75 - 34 * q^76 - 4 * q^77 + 9 * q^78 + 2 * q^79 - 45 * q^80 + 43 * q^81 - 18 * q^82 + 46 * q^83 - 103 * q^84 - 50 * q^85 + 78 * q^86 - 9 * q^87 - 24 * q^88 + 52 * q^89 - 46 * q^90 + 38 * q^91 + 54 * q^92 + 4 * q^93 + 3 * q^94 + 70 * q^95 - 21 * q^96 + 3 * q^97 - 120 * q^98 - 45 * 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 $

$2$	0.262885	0.185888	0.0929438	−	0.995671i	$-0.470372\pi$
$2$	0.262885	0.185888	0.0929438	+	0.995671i	$0.470372\pi$
$3$	0.849379	0.490389	0.245194	−	0.969474i	$-0.421148\pi$
$3$	0.849379	0.490389	0.245194	+	0.969474i	$0.421148\pi$
$4$	−1.93089	−0.965446
$5$	−1.83441	−0.820374	−0.410187	−	0.912001i	$-0.634537\pi$
$5$	−1.83441	−0.820374	−0.410187	+	0.912001i	$0.634537\pi$
$6$	0.223289	0.0911572
$7$	−0.312976	−0.118294	−0.0591469	−	0.998249i	$-0.518838\pi$
$7$	−0.312976	−0.118294	−0.0591469	+	0.998249i	$0.518838\pi$
$8$	−1.03337	−0.365352
$9$	−2.27856	−0.759519
$10$	−0.482239	−0.152497
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.64006	−0.473444
$13$	−1.17709	−0.326467	−0.163234	−	0.986587i	$-0.552192\pi$
$13$	−1.17709	−0.326467	−0.163234	+	0.986587i	$0.552192\pi$
$14$	−0.0822766	−0.0219893
$15$	−1.55811	−0.402302
$16$	3.59013	0.897531
$17$	2.54628	0.617565	0.308782	−	0.951133i	$-0.400079\pi$
$17$	2.54628	0.617565	0.308782	+	0.951133i	$0.400079\pi$
$18$	−0.598997	−0.141185
$19$	7.32694	1.68092	0.840458	−	0.541877i	$-0.182286\pi$
$19$	7.32694	1.68092	0.840458	+	0.541877i	$0.182286\pi$
$20$	3.54205	0.792027
$21$	−0.265835	−0.0580100
$22$	−0.262885	−0.0560472
$23$	−0.146051	−0.0304538	−0.0152269	−	0.999884i	$-0.504847\pi$
$23$	−0.146051	−0.0304538	−0.0152269	+	0.999884i	$0.504847\pi$
$24$	−0.877723	−0.179165
$25$	−1.63493	−0.326987
$26$	−0.309440	−0.0606862
$27$	−4.48349	−0.862849
$28$	0.604323	0.114206
$29$	4.58112	0.850692	0.425346	−	0.905031i	$-0.360152\pi$
$29$	4.58112	0.850692	0.425346	+	0.905031i	$0.360152\pi$
$30$	−0.409603	−0.0747830
$31$	0.837277	0.150379	0.0751897	−	0.997169i	$-0.476044\pi$
$31$	0.837277	0.150379	0.0751897	+	0.997169i	$0.476044\pi$
$32$	3.01053	0.532192
$33$	−0.849379	−0.147858
$34$	0.669379	0.114798
$35$	0.574127	0.0970451
$36$	4.39964	0.733274
$37$	4.83508	0.794883	0.397441	−	0.917628i	$-0.369898\pi$
$37$	4.83508	0.794883	0.397441	+	0.917628i	$0.369898\pi$
$38$	1.92614	0.312461
$39$	−0.999799	−0.160096
$40$	1.89563	0.299725
$41$	10.1987	1.59277	0.796383	−	0.604793i	$-0.206744\pi$
$41$	10.1987	1.59277	0.796383	+	0.604793i	$0.206744\pi$
$42$	−0.0698840	−0.0107833
$43$	8.06168	1.22939	0.614697	−	0.788763i	$-0.289278\pi$
$43$	8.06168	1.22939	0.614697	+	0.788763i	$0.289278\pi$
$44$	1.93089	0.291093
$45$	4.17981	0.623089
$46$	−0.0383946	−0.00566098
$47$	−2.40165	−0.350316	−0.175158	−	0.984540i	$-0.556044\pi$
$47$	−2.40165	−0.350316	−0.175158	+	0.984540i	$0.556044\pi$
$48$	3.04938	0.440140
$49$	−6.90205	−0.986007
$50$	−0.429799	−0.0607827
$51$	2.16276	0.302847
$52$	2.27284	0.315187
$53$	4.93783	0.678264	0.339132	−	0.940739i	$-0.389867\pi$
$53$	4.93783	0.678264	0.339132	+	0.940739i	$0.389867\pi$
$54$	−1.17864	−0.160393
$55$	1.83441	0.247352
$56$	0.323420	0.0432189
$57$	6.22335	0.824303
$58$	1.20431	0.158133
$59$	12.2163	1.59043	0.795217	−	0.606325i	$-0.207357\pi$
$59$	12.2163	1.59043	0.795217	+	0.606325i	$0.207357\pi$
$60$	3.00854	0.388401
$61$	−7.24820	−0.928037	−0.464018	−	0.885826i	$-0.653593\pi$
$61$	−7.24820	−0.928037	−0.464018	+	0.885826i	$0.653593\pi$
$62$	0.220107	0.0279537
$63$	0.713133	0.0898463
$64$	−6.38883	−0.798604
$65$	2.15928	0.267825
$66$	−0.223289	−0.0274849
$67$	−11.4402	−1.39764	−0.698821	−	0.715297i	$-0.746291\pi$
$67$	−11.4402	−1.39764	−0.698821	+	0.715297i	$0.746291\pi$
$68$	−4.91660	−0.596225
$69$	−0.124053	−0.0149342
$70$	0.150929	0.0180395
$71$	−3.06764	−0.364062	−0.182031	−	0.983293i	$-0.558267\pi$
$71$	−3.06764	−0.364062	−0.182031	+	0.983293i	$0.558267\pi$
$72$	2.35459	0.277492
$73$	3.48730	0.408157	0.204079	−	0.978955i	$-0.434580\pi$
$73$	3.48730	0.408157	0.204079	+	0.978955i	$0.434580\pi$
$74$	1.27107	0.147759
$75$	−1.38868	−0.160351
$76$	−14.1475	−1.62283
$77$	0.312976	0.0356669
$78$	−0.262832	−0.0297599
$79$	10.4102	1.17124	0.585618	−	0.810587i	$-0.300852\pi$
$79$	10.4102	1.17124	0.585618	+	0.810587i	$0.300852\pi$
$80$	−6.58577	−0.736311
$81$	3.02749	0.336387
$82$	2.68108	0.296075
$83$	−4.53584	−0.497873	−0.248936	−	0.968520i	$-0.580081\pi$
$83$	−4.53584	−0.497873	−0.248936	+	0.968520i	$0.580081\pi$
$84$	0.513299	0.0560055
$85$	−4.67093	−0.506634
$86$	2.11929	0.228529
$87$	3.89110	0.417170
$88$	1.03337	0.110158
$89$	9.16636	0.971632	0.485816	−	0.874061i	$-0.338522\pi$
$89$	9.16636	0.971632	0.485816	+	0.874061i	$0.338522\pi$
$90$	1.09881	0.115825
$91$	0.368402	0.0386191
$92$	0.282009	0.0294015
$93$	0.711165	0.0737444
$94$	−0.631356	−0.0651194
$95$	−13.4406	−1.37898
$96$	2.55708	0.260981
$97$	−10.4679	−1.06286	−0.531429	−	0.847103i	$-0.678345\pi$
$97$	−10.4679	−1.06286	−0.531429	+	0.847103i	$0.678345\pi$
$98$	−1.81444	−0.183286
$99$	2.27856	0.229003
$100$	3.15688	0.315688
$101$	4.88275	0.485851	0.242926	−	0.970045i	$-0.421893\pi$
$101$	4.88275	0.485851	0.242926	+	0.970045i	$0.421893\pi$
$102$	0.568556	0.0562955
$103$	6.55487	0.645870	0.322935	−	0.946421i	$-0.395330\pi$
$103$	6.55487	0.645870	0.322935	+	0.946421i	$0.395330\pi$
$104$	1.21638	0.119275
$105$	0.487651	0.0475899
$106$	1.29808	0.126081
$107$	−13.9535	−1.34893	−0.674467	−	0.738305i	$-0.735627\pi$
$107$	−13.9535	−1.34893	−0.674467	+	0.738305i	$0.735627\pi$
$108$	8.65714	0.833034
$109$	19.2812	1.84680	0.923400	−	0.383840i	$-0.125399\pi$
$109$	19.2812	1.84680	0.923400	+	0.383840i	$0.125399\pi$
$110$	0.482239	0.0459797
$111$	4.10682	0.389802
$112$	−1.12362	−0.106172
$113$	4.22660	0.397605	0.198802	−	0.980040i	$-0.436295\pi$
$113$	4.22660	0.397605	0.198802	+	0.980040i	$0.436295\pi$
$114$	1.63602	0.153228
$115$	0.267918	0.0249835
$116$	−8.84564	−0.821297
$117$	2.68208	0.247958
$118$	3.21149	0.295642
$119$	−0.796926	−0.0730541
$120$	1.61011	0.146982
$121$	1.00000	0.0909091
$122$	−1.90544	−0.172511
$123$	8.66253	0.781074
$124$	−1.61669	−0.145183
$125$	12.1712	1.08863
$126$	0.187472	0.0167013
$127$	15.5961	1.38393	0.691964	−	0.721932i	$-0.256746\pi$
$127$	15.5961	1.38393	0.691964	+	0.721932i	$0.256746\pi$
$128$	−7.70059	−0.680642
$129$	6.84742	0.602881
$130$	0.567641	0.0497854
$131$	1.00000	0.0873704
$131$	1.00000	0.0873704
$132$	1.64006	0.142749
$133$	−2.29316	−0.198842
$134$	−3.00745	−0.259804
$135$	8.22457	0.707858
$136$	−2.63126	−0.225628
$137$	−16.1027	−1.37575	−0.687873	−	0.725831i	$-0.741455\pi$
$137$	−16.1027	−1.37575	−0.687873	+	0.725831i	$0.741455\pi$
$138$	−0.0326116	−0.00277608
$139$	−9.93686	−0.842833	−0.421417	−	0.906867i	$-0.638467\pi$
$139$	−9.93686	−0.842833	−0.421417	+	0.906867i	$0.638467\pi$
$140$	−1.10858	−0.0936918
$141$	−2.03991	−0.171791
$142$	−0.806436	−0.0676746
$143$	1.17709	0.0984336
$144$	−8.18030	−0.681692
$145$	−8.40366	−0.697886
$146$	0.916757	0.0758713
$147$	−5.86245	−0.483527
$148$	−9.33602	−0.767416
$149$	2.88867	0.236649	0.118324	−	0.992975i	$-0.462248\pi$
$149$	2.88867	0.236649	0.118324	+	0.992975i	$0.462248\pi$
$150$	−0.365062	−0.0298072
$151$	−2.74804	−0.223632	−0.111816	−	0.993729i	$-0.535667\pi$
$151$	−2.74804	−0.223632	−0.111816	+	0.993729i	$0.535667\pi$
$152$	−7.57145	−0.614126
$153$	−5.80185	−0.469052
$154$	0.0822766	0.00663004
$155$	−1.53591	−0.123367
$156$	1.93050	0.154564
$157$	−5.68498	−0.453711	−0.226855	−	0.973928i	$-0.572844\pi$
$157$	−5.68498	−0.453711	−0.226855	+	0.973928i	$0.572844\pi$
$158$	2.73667	0.217718
$159$	4.19409	0.332613
$160$	−5.52255	−0.436596
$161$	0.0457105	0.00360249
$162$	0.795880	0.0625302
$163$	−15.3133	−1.19943	−0.599714	−	0.800214i	$-0.704719\pi$
$163$	−15.3133	−1.19943	−0.599714	+	0.800214i	$0.704719\pi$
$164$	−19.6925	−1.53773
$165$	1.55811	0.121299
$166$	−1.19240	−0.0925484
$167$	11.9125	0.921815	0.460908	−	0.887448i	$-0.347524\pi$
$167$	11.9125	0.921815	0.460908	+	0.887448i	$0.347524\pi$
$168$	0.274706	0.0211941
$169$	−11.6144	−0.893419
$170$	−1.22792	−0.0941769
$171$	−16.6949	−1.27669
$172$	−15.5662	−1.18691
$173$	22.4862	1.70959	0.854796	−	0.518964i	$-0.173682\pi$
$173$	22.4862	1.70959	0.854796	+	0.518964i	$0.173682\pi$
$174$	1.02291	0.0775467
$175$	0.511695	0.0386805
$176$	−3.59013	−0.270616
$177$	10.3763	0.779931
$178$	2.40970	0.180614
$179$	2.84666	0.212770	0.106385	−	0.994325i	$-0.466072\pi$
$179$	2.84666	0.212770	0.106385	+	0.994325i	$0.466072\pi$
$180$	−8.07076	−0.601559
$181$	17.2818	1.28454	0.642272	−	0.766477i	$-0.277992\pi$
$181$	17.2818	1.28454	0.642272	+	0.766477i	$0.277992\pi$
$182$	0.0968473	0.00717880
$183$	−6.15647	−0.455099
$184$	0.150925	0.0111263
$185$	−8.86953	−0.652101
$186$	0.186954	0.0137082
$187$	−2.54628	−0.186203
$188$	4.63732	0.338211
$189$	1.40323	0.102070
$190$	−3.53334	−0.256335
$191$	−8.72950	−0.631645	−0.315822	−	0.948818i	$-0.602280\pi$
$191$	−8.72950	−0.631645	−0.315822	+	0.948818i	$0.602280\pi$
$192$	−5.42653	−0.391626
$193$	−23.2570	−1.67408	−0.837038	−	0.547144i	$-0.815715\pi$
$193$	−23.2570	−1.67408	−0.837038	+	0.547144i	$0.815715\pi$
$194$	−2.75186	−0.197572
$195$	1.83404	0.131339
$196$	13.3271	0.951936
$197$	26.2549	1.87058	0.935292	−	0.353878i	$-0.115137\pi$
$197$	26.2549	1.87058	0.935292	+	0.353878i	$0.115137\pi$
$198$	0.598997	0.0425689
$199$	9.05650	0.641998	0.320999	−	0.947079i	$-0.395981\pi$
$199$	9.05650	0.641998	0.320999	+	0.947079i	$0.395981\pi$
$200$	1.68949	0.119465
$201$	−9.71705	−0.685388
$202$	1.28360	0.0903137
$203$	−1.43378	−0.100632
$204$	−4.17605	−0.292382
$205$	−18.7086	−1.30666
$206$	1.72317	0.120059
$207$	0.332786	0.0231302
$208$	−4.22592	−0.293015
$209$	−7.32694	−0.506815
$210$	0.128196	0.00884636
$211$	16.4296	1.13106	0.565531	−	0.824727i	$-0.308671\pi$
$211$	16.4296	1.13106	0.565531	+	0.824727i	$0.308671\pi$
$212$	−9.53442	−0.654827
$213$	−2.60559	−0.178532
$214$	−3.66816	−0.250750
$215$	−14.7884	−1.00856
$216$	4.63311	0.315243
$217$	−0.262048	−0.0177889
$218$	5.06872	0.343297
$219$	2.96203	0.200156
$220$	−3.54205	−0.238805
$221$	−2.99722	−0.201615
$222$	1.07962	0.0724593
$223$	−7.85435	−0.525966	−0.262983	−	0.964800i	$-0.584706\pi$
$223$	−7.85435	−0.525966	−0.262983	+	0.964800i	$0.584706\pi$
$224$	−0.942224	−0.0629550
$225$	3.72529	0.248352
$226$	1.11111	0.0739098
$227$	−7.35783	−0.488356	−0.244178	−	0.969730i	$-0.578518\pi$
$227$	−7.35783	−0.488356	−0.244178	+	0.969730i	$0.578518\pi$
$228$	−12.0166	−0.795820
$229$	23.6322	1.56166	0.780831	−	0.624742i	$-0.214796\pi$
$229$	23.6322	1.56166	0.780831	+	0.624742i	$0.214796\pi$
$230$	0.0704315	0.00464412
$231$	0.265835	0.0174907
$232$	−4.73399	−0.310802
$233$	14.3537	0.940343	0.470172	−	0.882575i	$-0.344192\pi$
$233$	14.3537	0.940343	0.470172	+	0.882575i	$0.344192\pi$
$234$	0.705077	0.0460923
$235$	4.40561	0.287390
$236$	−23.5884	−1.53548
$237$	8.84217	0.574361
$238$	−0.209500	−0.0135798
$239$	−7.44969	−0.481881	−0.240940	−	0.970540i	$-0.577456\pi$
$239$	−7.44969	−0.481881	−0.240940	+	0.970540i	$0.577456\pi$
$240$	−5.59381	−0.361079
$241$	9.24873	0.595763	0.297881	−	0.954603i	$-0.403720\pi$
$241$	9.24873	0.595763	0.297881	+	0.954603i	$0.403720\pi$
$242$	0.262885	0.0168989
$243$	16.0220	1.02781
$244$	13.9955	0.895969
$245$	12.6612	0.808894
$246$	2.27725	0.145192
$247$	−8.62451	−0.548764
$248$	−0.865218	−0.0549414
$249$	−3.85264	−0.244151
$250$	3.19962	0.202362
$251$	12.7298	0.803499	0.401750	−	0.915750i	$-0.368402\pi$
$251$	12.7298	0.803499	0.401750	+	0.915750i	$0.368402\pi$
$252$	−1.37698	−0.0867418
$253$	0.146051	0.00918216
$254$	4.09997	0.257255
$255$	−3.96739	−0.248448
$256$	10.7533	0.672081
$257$	10.6433	0.663909	0.331954	−	0.943295i	$-0.392292\pi$
$257$	10.6433	0.663909	0.331954	+	0.943295i	$0.392292\pi$
$258$	1.80008	0.112068
$259$	−1.51326	−0.0940297
$260$	−4.16933	−0.258571
$261$	−10.4383	−0.646117
$262$	0.262885	0.0162411
$263$	13.3361	0.822340	0.411170	−	0.911559i	$-0.365120\pi$
$263$	13.3361	0.822340	0.411170	+	0.911559i	$0.365120\pi$
$264$	0.877723	0.0540201
$265$	−9.05802	−0.556430
$266$	−0.602836	−0.0369622
$267$	7.78571	0.476478
$268$	22.0898	1.34935
$269$	−21.4748	−1.30934	−0.654671	−	0.755914i	$-0.727193\pi$
$269$	−21.4748	−1.30934	−0.654671	+	0.755914i	$0.727193\pi$
$270$	2.16211	0.131582
$271$	−26.3662	−1.60164	−0.800818	−	0.598908i	$-0.795601\pi$
$271$	−26.3662	−1.60164	−0.800818	+	0.598908i	$0.795601\pi$
$272$	9.14148	0.554284
$273$	0.312913	0.0189384
$274$	−4.23315	−0.255734
$275$	1.63493	0.0985902
$276$	0.239532	0.0144182
$277$	−19.6986	−1.18358	−0.591789	−	0.806093i	$-0.701578\pi$
$277$	−19.6986	−1.18358	−0.591789	+	0.806093i	$0.701578\pi$
$278$	−2.61225	−0.156672
$279$	−1.90778	−0.114216
$280$	−0.593286	−0.0354556
$281$	1.26938	0.0757246	0.0378623	−	0.999283i	$-0.487945\pi$
$281$	1.26938	0.0757246	0.0378623	+	0.999283i	$0.487945\pi$
$282$	−0.536260	−0.0319338
$283$	31.0399	1.84513	0.922566	−	0.385840i	$-0.126088\pi$
$283$	31.0399	1.84513	0.922566	+	0.385840i	$0.126088\pi$
$284$	5.92328	0.351482
$285$	−11.4162	−0.676236
$286$	0.309440	0.0182976
$287$	−3.19194	−0.188414
$288$	−6.85966	−0.404210
$289$	−10.5164	−0.618614
$290$	−2.20919	−0.129728
$291$	−8.89125	−0.521214
$292$	−6.73359	−0.394054
$293$	−15.5995	−0.911333	−0.455667	−	0.890150i	$-0.650599\pi$
$293$	−15.5995	−0.911333	−0.455667	+	0.890150i	$0.650599\pi$
$294$	−1.54115	−0.0898816
$295$	−22.4098	−1.30475
$296$	−4.99643	−0.290412
$297$	4.48349	0.260159
$298$	0.759387	0.0439901
$299$	0.171916	0.00994216
$300$	2.68139	0.154810
$301$	−2.52311	−0.145430
$302$	−0.722418	−0.0415705
$303$	4.14730	0.238256
$304$	26.3047	1.50868
$305$	13.2962	0.761337
$306$	−1.52522	−0.0871909
$307$	−3.00385	−0.171439	−0.0857195	−	0.996319i	$-0.527319\pi$
$307$	−3.00385	−0.171439	−0.0857195	+	0.996319i	$0.527319\pi$
$308$	−0.604323	−0.0344345
$309$	5.56756	0.316728
$310$	−0.403767	−0.0229325
$311$	−28.9169	−1.63972	−0.819862	−	0.572561i	$-0.805950\pi$
$311$	−28.9169	−1.63972	−0.819862	+	0.572561i	$0.805950\pi$
$312$	1.03316	0.0584914
$313$	19.5870	1.10712	0.553561	−	0.832809i	$-0.313269\pi$
$313$	19.5870	1.10712	0.553561	+	0.832809i	$0.313269\pi$
$314$	−1.49449	−0.0843392
$315$	−1.30818	−0.0737076
$316$	−20.1009	−1.13076
$317$	−17.1234	−0.961746	−0.480873	−	0.876790i	$-0.659680\pi$
$317$	−17.1234	−0.961746	−0.480873	+	0.876790i	$0.659680\pi$
$318$	1.10256	0.0618286
$319$	−4.58112	−0.256493
$320$	11.7197	0.655154
$321$	−11.8518	−0.661503
$322$	0.0120166	0.000669658 0
$323$	18.6565	1.03807
$324$	−5.84575	−0.324764
$325$	1.92447	0.106750
$326$	−4.02562	−0.222959
$327$	16.3770	0.905650
$328$	−10.5390	−0.581920
$329$	0.751657	0.0414402
$330$	0.409603	0.0225479
$331$	−10.6532	−0.585554	−0.292777	−	0.956181i	$-0.594579\pi$
$331$	−10.6532	−0.585554	−0.292777	+	0.956181i	$0.594579\pi$
$332$	8.75821	0.480669
$333$	−11.0170	−0.603728
$334$	3.13161	0.171354
$335$	20.9860	1.14659
$336$	−0.954381	−0.0520658
$337$	−31.1033	−1.69431	−0.847153	−	0.531349i	$-0.821685\pi$
$337$	−31.1033	−1.69431	−0.847153	+	0.531349i	$0.821685\pi$
$338$	−3.05326	−0.166075
$339$	3.58998	0.194981
$340$	9.01907	0.489128
$341$	−0.837277	−0.0453411
$342$	−4.38882	−0.237320
$343$	4.35101	0.234932
$344$	−8.33070	−0.449161
$345$	0.227564	0.0122516
$346$	5.91127	0.317792
$347$	14.9718	0.803726	0.401863	−	0.915700i	$-0.368363\pi$
$347$	14.9718	0.803726	0.401863	+	0.915700i	$0.368363\pi$
$348$	−7.51330	−0.402755
$349$	2.90510	0.155507	0.0777533	−	0.996973i	$-0.475225\pi$
$349$	2.90510	0.155507	0.0777533	+	0.996973i	$0.475225\pi$
$350$	0.134517	0.00719022
$351$	5.27750	0.281692
$352$	−3.01053	−0.160462
$353$	35.6745	1.89876	0.949380	−	0.314130i	$-0.101713\pi$
$353$	35.6745	1.89876	0.949380	+	0.314130i	$0.101713\pi$
$354$	2.72777	0.144979
$355$	5.62731	0.298667
$356$	−17.6993	−0.938059
$357$	−0.676892	−0.0358249
$358$	0.748344	0.0395512
$359$	30.3501	1.60182	0.800908	−	0.598787i	$-0.204351\pi$
$359$	30.3501	1.60182	0.800908	+	0.598787i	$0.204351\pi$
$360$	−4.31930	−0.227647
$361$	34.6841	1.82548
$362$	4.54311	0.238781
$363$	0.849379	0.0445808
$364$	−0.711345	−0.0372846
$365$	−6.39714	−0.334841
$366$	−1.61844	−0.0845973
$367$	29.2773	1.52826	0.764130	−	0.645062i	$-0.223169\pi$
$367$	29.2773	1.52826	0.764130	+	0.645062i	$0.223169\pi$
$368$	−0.524342	−0.0273332
$369$	−23.2382	−1.20973
$370$	−2.33166	−0.121217
$371$	−1.54542	−0.0802344
$372$	−1.37318	−0.0711962
$373$	6.41327	0.332067	0.166033	−	0.986120i	$-0.446904\pi$
$373$	6.41327	0.332067	0.166033	+	0.986120i	$0.446904\pi$
$374$	−0.669379	−0.0346128
$375$	10.3380	0.533850
$376$	2.48179	0.127989
$377$	−5.39241	−0.277723
$378$	0.368886	0.0189735
$379$	−31.8650	−1.63679	−0.818397	−	0.574653i	$-0.805137\pi$
$379$	−31.8650	−1.63679	−0.818397	+	0.574653i	$0.805137\pi$
$380$	25.9524	1.33133
$381$	13.2470	0.678663
$382$	−2.29485	−0.117415
$383$	10.8251	0.553134	0.276567	−	0.960995i	$-0.410803\pi$
$383$	10.8251	0.553134	0.276567	+	0.960995i	$0.410803\pi$
$384$	−6.54072	−0.333779
$385$	−0.574127	−0.0292602
$386$	−6.11391	−0.311190
$387$	−18.3690	−0.933748
$388$	20.2125	1.02613
$389$	−32.4137	−1.64344	−0.821720	−	0.569891i	$-0.806985\pi$
$389$	−32.4137	−1.64344	−0.821720	+	0.569891i	$0.806985\pi$
$390$	0.482142	0.0244142
$391$	−0.371888	−0.0188072
$392$	7.13238	0.360239
$393$	0.849379	0.0428455
$394$	6.90201	0.347718
$395$	−19.0965	−0.960851
$396$	−4.39964	−0.221090
$397$	−28.6658	−1.43869	−0.719347	−	0.694651i	$-0.755559\pi$
$397$	−28.6658	−1.43869	−0.719347	+	0.694651i	$0.755559\pi$
$398$	2.38082	0.119339
$399$	−1.94776	−0.0975099
$400$	−5.86962	−0.293481
$401$	−6.36305	−0.317755	−0.158878	−	0.987298i	$-0.550788\pi$
$401$	−6.36305	−0.317755	−0.158878	+	0.987298i	$0.550788\pi$
$402$	−2.55446	−0.127405
$403$	−0.985554	−0.0490940
$404$	−9.42806	−0.469063
$405$	−5.55366	−0.275963
$406$	−0.376919	−0.0187062
$407$	−4.83508	−0.239666
$408$	−2.23493	−0.110646
$409$	−31.6698	−1.56597	−0.782985	−	0.622040i	$-0.786304\pi$
$409$	−31.6698	−1.56597	−0.782985	+	0.622040i	$0.786304\pi$
$410$	−4.91820	−0.242892
$411$	−13.6773	−0.674651
$412$	−12.6567	−0.623553
$413$	−3.82342	−0.188138
$414$	0.0874843	0.00429962
$415$	8.32060	0.408442
$416$	−3.54368	−0.173743
$417$	−8.44016	−0.413316
$418$	−1.92614	−0.0942107
$419$	16.6142	0.811656	0.405828	−	0.913949i	$-0.366983\pi$
$419$	16.6142	0.811656	0.405828	+	0.913949i	$0.366983\pi$
$420$	−0.941601	−0.0459454
$421$	12.4544	0.606991	0.303495	−	0.952833i	$-0.401846\pi$
$421$	12.4544	0.606991	0.303495	+	0.952833i	$0.401846\pi$
$422$	4.31910	0.210250
$423$	5.47228	0.266072
$424$	−5.10261	−0.247805
$425$	−4.16300	−0.201935
$426$	−0.684969	−0.0331869
$427$	2.26851	0.109781
$428$	26.9427	1.30232
$429$	0.999799	0.0482708
$430$	−3.88765	−0.187479
$431$	2.87456	0.138463	0.0692313	−	0.997601i	$-0.477945\pi$
$431$	2.87456	0.138463	0.0692313	+	0.997601i	$0.477945\pi$
$432$	−16.0963	−0.774434
$433$	41.2104	1.98044	0.990222	−	0.139502i	$-0.0445503\pi$
$433$	41.2104	1.98044	0.990222	+	0.139502i	$0.0445503\pi$
$434$	−0.0688883	−0.00330674
$435$	−7.13789	−0.342235
$436$	−37.2298	−1.78298
$437$	−1.07011	−0.0511902
$438$	0.778673	0.0372065
$439$	−7.56872	−0.361235	−0.180618	−	0.983553i	$-0.557810\pi$
$439$	−7.56872	−0.361235	−0.180618	+	0.983553i	$0.557810\pi$
$440$	−1.89563	−0.0903705
$441$	15.7267	0.748890
$442$	−0.787923	−0.0374777
$443$	22.7664	1.08166	0.540832	−	0.841131i	$-0.318109\pi$
$443$	22.7664	1.08166	0.540832	+	0.841131i	$0.318109\pi$
$444$	−7.92982	−0.376332
$445$	−16.8149	−0.797102
$446$	−2.06479	−0.0977705
$447$	2.45357	0.116050
$448$	1.99955	0.0944698
$449$	36.7737	1.73546	0.867730	−	0.497036i	$-0.165578\pi$
$449$	36.7737	1.73546	0.867730	+	0.497036i	$0.165578\pi$
$450$	0.979321	0.0461656
$451$	−10.1987	−0.480237
$452$	−8.16110	−0.383866
$453$	−2.33413	−0.109667
$454$	−1.93426	−0.0907793
$455$	−0.675802	−0.0316821
$456$	−6.43103	−0.301161
$457$	−26.6601	−1.24711	−0.623553	−	0.781781i	$-0.714312\pi$
$457$	−26.6601	−1.24711	−0.623553	+	0.781781i	$0.714312\pi$
$458$	6.21256	0.290294
$459$	−11.4162	−0.532865
$460$	−0.517321	−0.0241202
$461$	−15.6022	−0.726668	−0.363334	−	0.931659i	$-0.618362\pi$
$461$	−15.6022	−0.726668	−0.363334	+	0.931659i	$0.618362\pi$
$462$	0.0698840	0.00325130
$463$	13.7009	0.636737	0.318368	−	0.947967i	$-0.396865\pi$
$463$	13.7009	0.636737	0.318368	+	0.947967i	$0.396865\pi$
$464$	16.4468	0.763523
$465$	−1.30457	−0.0604980
$466$	3.77337	0.174798
$467$	−17.7358	−0.820715	−0.410357	−	0.911925i	$-0.634596\pi$
$467$	−17.7358	−0.820715	−0.410357	+	0.911925i	$0.634596\pi$
$468$	−5.17880	−0.239390
$469$	3.58050	0.165332
$470$	1.15817	0.0534222
$471$	−4.82870	−0.222495
$472$	−12.6240	−0.581068
$473$	−8.06168	−0.370676
$474$	2.32447	0.106767
$475$	−11.9791	−0.549637
$476$	1.53878	0.0705297
$477$	−11.2511	−0.515154
$478$	−1.95841	−0.0895756
$479$	−4.24333	−0.193883	−0.0969413	−	0.995290i	$-0.530906\pi$
$479$	−4.24333	−0.193883	−0.0969413	+	0.995290i	$0.530906\pi$
$480$	−4.69074	−0.214102
$481$	−5.69135	−0.259503
$482$	2.43135	0.110745
$483$	0.0388255	0.00176662
$484$	−1.93089	−0.0877678
$485$	19.2025	0.871942
$486$	4.21193	0.191057
$487$	−9.23744	−0.418588	−0.209294	−	0.977853i	$-0.567117\pi$
$487$	−9.23744	−0.418588	−0.209294	+	0.977853i	$0.567117\pi$
$488$	7.49008	0.339060
$489$	−13.0068	−0.588186
$490$	3.32843	0.150363
$491$	5.86234	0.264564	0.132282	−	0.991212i	$-0.457770\pi$
$491$	5.86234	0.264564	0.132282	+	0.991212i	$0.457770\pi$
$492$	−16.7264	−0.754085
$493$	11.6648	0.525357
$494$	−2.26725	−0.102008
$495$	−4.17981	−0.187868
$496$	3.00593	0.134970
$497$	0.960097	0.0430663
$498$	−1.01280	−0.0453847
$499$	19.6621	0.880195	0.440098	−	0.897950i	$-0.354944\pi$
$499$	19.6621	0.880195	0.440098	+	0.897950i	$0.354944\pi$
$500$	−23.5013	−1.05101
$501$	10.1182	0.452048
$502$	3.34647	0.149360
$503$	−23.9809	−1.06926	−0.534628	−	0.845087i	$-0.679548\pi$
$503$	−23.9809	−1.06926	−0.534628	+	0.845087i	$0.679548\pi$
$504$	−0.736931	−0.0328255
$505$	−8.95697	−0.398580
$506$	0.0383946	0.00170685
$507$	−9.86506	−0.438123
$508$	−30.1143	−1.33611
$509$	36.6789	1.62576	0.812882	−	0.582428i	$-0.197897\pi$
$509$	36.6789	1.62576	0.812882	+	0.582428i	$0.197897\pi$
$510$	−1.04297	−0.0461833
$511$	−1.09144	−0.0482824
$512$	18.2281	0.805574
$513$	−32.8503	−1.45038
$514$	2.79795	0.123412
$515$	−12.0243	−0.529855
$516$	−13.2216	−0.582049
$517$	2.40165	0.105624
$518$	−0.397814	−0.0174789
$519$	19.0993	0.838365
$520$	−2.23133	−0.0978505
$521$	9.23790	0.404720	0.202360	−	0.979311i	$-0.435139\pi$
$521$	9.23790	0.404720	0.202360	+	0.979311i	$0.435139\pi$
$522$	−2.74408	−0.120105
$523$	22.7675	0.995552	0.497776	−	0.867306i	$-0.334150\pi$
$523$	22.7675	0.995552	0.497776	+	0.867306i	$0.334150\pi$
$524$	−1.93089	−0.0843514
$525$	0.434623	0.0189685
$526$	3.50586	0.152863
$527$	2.13195	0.0928690
$528$	−3.04938	−0.132707
$529$	−22.9787	−0.999073
$530$	−2.38121	−0.103433
$531$	−27.8356	−1.20796
$532$	4.42784	0.191971
$533$	−12.0048	−0.519986
$534$	2.04674	0.0885713
$535$	25.5964	1.10663
$536$	11.8220	0.510631
$537$	2.41790	0.104340
$538$	−5.64540	−0.243390
$539$	6.90205	0.297292
$540$	−15.8808	−0.683399
$541$	22.3591	0.961291	0.480646	−	0.876915i	$-0.340402\pi$
$541$	22.3591	0.961291	0.480646	+	0.876915i	$0.340402\pi$
$542$	−6.93128	−0.297724
$543$	14.6788	0.629926
$544$	7.66567	0.328663
$545$	−35.3696	−1.51507
$546$	0.0822601	0.00352041
$547$	−0.0188390	−0.000805496 0	−0.000402748	−	1.00000i	$-0.500128\pi$
$547$	−0.0188390	−0.000805496 0	−0.000402748		1.00000i	$0.500128\pi$
$548$	31.0926	1.32821
$549$	16.5154	0.704861
$550$	0.429799	0.0183267
$551$	33.5656	1.42994
$552$	0.128193	0.00545624
$553$	−3.25813	−0.138550
$554$	−5.17847	−0.220012
$555$	−7.53359	−0.319783
$556$	19.1870	0.813710
$557$	23.2665	0.985834	0.492917	−	0.870076i	$-0.335931\pi$
$557$	23.2665	0.985834	0.492917	+	0.870076i	$0.335931\pi$
$558$	−0.501527	−0.0212313
$559$	−9.48936	−0.401357
$560$	2.06119	0.0871011
$561$	−2.16276	−0.0913118
$562$	0.333700	0.0140763
$563$	−19.0275	−0.801916	−0.400958	−	0.916097i	$-0.631323\pi$
$563$	−19.0275	−0.801916	−0.400958	+	0.916097i	$0.631323\pi$
$564$	3.93884	0.165855
$565$	−7.75332	−0.326185
$566$	8.15992	0.342987
$567$	−0.947530	−0.0397925
$568$	3.17001	0.133011
$569$	9.64732	0.404437	0.202218	−	0.979340i	$-0.435185\pi$
$569$	9.64732	0.404437	0.202218	+	0.979340i	$0.435185\pi$
$570$	−3.00114	−0.125704
$571$	−44.8915	−1.87865	−0.939325	−	0.343028i	$-0.888548\pi$
$571$	−44.8915	−1.87865	−0.939325	+	0.343028i	$0.888548\pi$
$572$	−2.27284	−0.0950323
$573$	−7.41465	−0.309752
$574$	−0.839112	−0.0350239
$575$	0.238784	0.00995798
$576$	14.5573	0.606554
$577$	−17.1091	−0.712263	−0.356131	−	0.934436i	$-0.615904\pi$
$577$	−17.1091	−0.712263	−0.356131	+	0.934436i	$0.615904\pi$
$578$	−2.76461	−0.114993
$579$	−19.7540	−0.820949
$580$	16.2265	0.673771
$581$	1.41961	0.0588953
$582$	−2.33737	−0.0968872
$583$	−4.93783	−0.204504
$584$	−3.60367	−0.149121
$585$	−4.92003	−0.203418
$586$	−4.10087	−0.169406
$587$	0.959298	0.0395945	0.0197972	−	0.999804i	$-0.493698\pi$
$587$	0.959298	0.0395945	0.0197972	+	0.999804i	$0.493698\pi$
$588$	11.3198	0.466819
$589$	6.13468	0.252775
$590$	−5.89120	−0.242537
$591$	22.3003	0.917313
$592$	17.3586	0.713432
$593$	16.7627	0.688362	0.344181	−	0.938903i	$-0.388157\pi$
$593$	16.7627	0.688362	0.344181	+	0.938903i	$0.388157\pi$
$594$	1.17864	0.0483602
$595$	1.46189	0.0599316
$596$	−5.57770	−0.228472
$597$	7.69240	0.314829
$598$	0.0451941	0.00184812
$599$	−8.33148	−0.340415	−0.170208	−	0.985408i	$-0.554444\pi$
$599$	−8.33148	−0.340415	−0.170208	+	0.985408i	$0.554444\pi$
$600$	1.43502	0.0585844
$601$	5.97229	0.243615	0.121807	−	0.992554i	$-0.461131\pi$
$601$	5.97229	0.243615	0.121807	+	0.992554i	$0.461131\pi$
$602$	−0.663287	−0.0270336
$603$	26.0671	1.06153
$604$	5.30617	0.215905
$605$	−1.83441	−0.0745794
$606$	1.09026	0.0442889
$607$	−3.72504	−0.151195	−0.0755974	−	0.997138i	$-0.524086\pi$
$607$	−3.72504	−0.151195	−0.0755974	+	0.997138i	$0.524086\pi$
$608$	22.0580	0.894570
$609$	−1.21782	−0.0493486
$610$	3.49536	0.141523
$611$	2.82696	0.114367
$612$	11.2027	0.452844
$613$	−10.7270	−0.433259	−0.216630	−	0.976254i	$-0.569506\pi$
$613$	−10.7270	−0.433259	−0.216630	+	0.976254i	$0.569506\pi$
$614$	−0.789667	−0.0318684
$615$	−15.8907	−0.640773
$616$	−0.323420	−0.0130310
$617$	9.64918	0.388462	0.194231	−	0.980956i	$-0.437779\pi$
$617$	9.64918	0.388462	0.194231	+	0.980956i	$0.437779\pi$
$618$	1.46363	0.0588757
$619$	31.3535	1.26020	0.630101	−	0.776513i	$-0.283013\pi$
$619$	31.3535	1.26020	0.630101	+	0.776513i	$0.283013\pi$
$620$	2.96568	0.119104
$621$	0.654819	0.0262770
$622$	−7.60180	−0.304804
$623$	−2.86885	−0.114938
$624$	−3.58940	−0.143691
$625$	−14.1523	−0.566093
$626$	5.14912	0.205800
$627$	−6.22335	−0.248537
$628$	10.9771	0.438033
$629$	12.3115	0.490891
$630$	−0.343900	−0.0137013
$631$	16.9788	0.675915	0.337957	−	0.941161i	$-0.390264\pi$
$631$	16.9788	0.675915	0.337957	+	0.941161i	$0.390264\pi$
$632$	−10.7576	−0.427913
$633$	13.9550	0.554660
$634$	−4.50148	−0.178777
$635$	−28.6096	−1.13534
$636$	−8.09833	−0.321120
$637$	8.12436	0.321899
$638$	−1.20431	−0.0476789
$639$	6.98979	0.276512
$640$	14.1261	0.558381
$641$	6.35120	0.250857	0.125429	−	0.992103i	$-0.459969\pi$
$641$	6.35120	0.250857	0.125429	+	0.992103i	$0.459969\pi$
$642$	−3.11566	−0.122965
$643$	13.7154	0.540884	0.270442	−	0.962736i	$-0.412830\pi$
$643$	13.7154	0.540884	0.270442	+	0.962736i	$0.412830\pi$
$644$	−0.0882620	−0.00347801
$645$	−12.5610	−0.494588
$646$	4.90450	0.192965
$647$	42.4155	1.66753	0.833763	−	0.552123i	$-0.186182\pi$
$647$	42.4155	1.66753	0.833763	+	0.552123i	$0.186182\pi$
$648$	−3.12852	−0.122900
$649$	−12.2163	−0.479534
$650$	0.505914	0.0198436
$651$	−0.222578	−0.00872350
$652$	29.5683	1.15798
$653$	1.81593	0.0710628	0.0355314	−	0.999369i	$-0.488688\pi$
$653$	1.81593	0.0710628	0.0355314	+	0.999369i	$0.488688\pi$
$654$	4.30526	0.168349
$655$	−1.83441	−0.0716764
$656$	36.6145	1.42956
$657$	−7.94600	−0.310003
$658$	0.197599	0.00770322
$659$	−0.817225	−0.0318346	−0.0159173	−	0.999873i	$-0.505067\pi$
$659$	−0.817225	−0.0318346	−0.0159173	+	0.999873i	$0.505067\pi$
$660$	−3.00854	−0.117107
$661$	−37.6156	−1.46308	−0.731539	−	0.681800i	$-0.761198\pi$
$661$	−37.6156	−1.46308	−0.731539	+	0.681800i	$0.761198\pi$
$662$	−2.80057	−0.108847
$663$	−2.54577	−0.0988696
$664$	4.68720	0.181899
$665$	4.20659	0.163125
$666$	−2.89620	−0.112226
$667$	−0.669078	−0.0259068
$668$	−23.0017	−0.889963
$669$	−6.67131	−0.257928
$670$	5.51690	0.213136
$671$	7.24820	0.279814
$672$	−0.800305	−0.0308724
$673$	16.6552	0.642013	0.321006	−	0.947077i	$-0.395979\pi$
$673$	16.6552	0.642013	0.321006	+	0.947077i	$0.395979\pi$
$674$	−8.17658	−0.314950
$675$	7.33021	0.282140
$676$	22.4262	0.862548
$677$	−1.70613	−0.0655721	−0.0327860	−	0.999462i	$-0.510438\pi$
$677$	−1.70613	−0.0655721	−0.0327860	+	0.999462i	$0.510438\pi$
$678$	0.943751	0.0362445
$679$	3.27622	0.125730
$680$	4.82681	0.185100
$681$	−6.24958	−0.239485
$682$	−0.220107	−0.00842834
$683$	−15.0946	−0.577577	−0.288789	−	0.957393i	$-0.593252\pi$
$683$	−15.0946	−0.577577	−0.288789	+	0.957393i	$0.593252\pi$
$684$	32.2360	1.23257
$685$	29.5390	1.12863
$686$	1.14381	0.0436710
$687$	20.0727	0.765822
$688$	28.9424	1.10342
$689$	−5.81230	−0.221431
$690$	0.0598230	0.00227742
$691$	1.90197	0.0723542	0.0361771	−	0.999345i	$-0.488482\pi$
$691$	1.90197	0.0723542	0.0361771	+	0.999345i	$0.488482\pi$
$692$	−43.4184	−1.65052
$693$	−0.713133	−0.0270897
$694$	3.93585	0.149403
$695$	18.2283	0.691439
$696$	−4.02095	−0.152414
$697$	25.9687	0.983635
$698$	0.763707	0.0289067
$699$	12.1917	0.461134
$700$	−0.988027	−0.0373439
$701$	−2.48508	−0.0938601	−0.0469301	−	0.998898i	$-0.514944\pi$
$701$	−2.48508	−0.0938601	−0.0469301	+	0.998898i	$0.514944\pi$
$702$	1.38737	0.0523630
$703$	35.4264	1.33613
$704$	6.38883	0.240788
$705$	3.74203	0.140933
$706$	9.37827	0.352956
$707$	−1.52818	−0.0574732
$708$	−20.0355	−0.752981
$709$	7.33626	0.275519	0.137760	−	0.990466i	$-0.456010\pi$
$709$	7.33626	0.275519	0.137760	+	0.990466i	$0.456010\pi$
$710$	1.47933	0.0555184
$711$	−23.7201	−0.889575
$712$	−9.47226	−0.354988
$713$	−0.122285	−0.00457962
$714$	−0.177944	−0.00665940
$715$	−2.15928	−0.0807524
$716$	−5.49660	−0.205418
$717$	−6.32761	−0.236309
$718$	7.97857	0.297758
$719$	51.2730	1.91216	0.956080	−	0.293105i	$-0.0946887\pi$
$719$	51.2730	1.91216	0.956080	+	0.293105i	$0.0946887\pi$
$720$	15.0060	0.559242
$721$	−2.05152	−0.0764024
$722$	9.11792	0.339334
$723$	7.85567	0.292155
$724$	−33.3692	−1.24016
$725$	−7.48982	−0.278165
$726$	0.223289	0.00828702
$727$	−38.9716	−1.44538	−0.722688	−	0.691175i	$-0.757094\pi$
$727$	−38.9716	−1.44538	−0.722688	+	0.691175i	$0.757094\pi$
$728$	−0.380696	−0.0141095
$729$	4.52625	0.167639
$730$	−1.68171	−0.0622428
$731$	20.5273	0.759230
$732$	11.8875	0.439374
$733$	−43.6941	−1.61388	−0.806939	−	0.590635i	$-0.798877\pi$
$733$	−43.6941	−1.61388	−0.806939	+	0.590635i	$0.798877\pi$
$734$	7.69654	0.284085
$735$	10.7541	0.396673
$736$	−0.439692	−0.0162072
$737$	11.4402	0.421405
$738$	−6.10898	−0.224875
$739$	−10.4423	−0.384125	−0.192062	−	0.981383i	$-0.561518\pi$
$739$	−10.4423	−0.384125	−0.192062	+	0.981383i	$0.561518\pi$
$740$	17.1261	0.629568
$741$	−7.32547	−0.269108
$742$	−0.406268	−0.0149146
$743$	27.3610	1.00378	0.501888	−	0.864933i	$-0.332639\pi$
$743$	27.3610	1.00378	0.501888	+	0.864933i	$0.332639\pi$
$744$	−0.734898	−0.0269427
$745$	−5.29901	−0.194141
$746$	1.68595	0.0617271
$747$	10.3352	0.378144
$748$	4.91660	0.179769
$749$	4.36711	0.159571
$750$	2.71769	0.0992360
$751$	11.9413	0.435744	0.217872	−	0.975977i	$-0.430089\pi$
$751$	11.9413	0.435744	0.217872	+	0.975977i	$0.430089\pi$
$752$	−8.62221	−0.314420
$753$	10.8124	0.394027
$754$	−1.41758	−0.0516253
$755$	5.04104	0.183462
$756$	−2.70948	−0.0985427
$757$	2.83378	0.102995	0.0514977	−	0.998673i	$-0.483601\pi$
$757$	2.83378	0.102995	0.0514977	+	0.998673i	$0.483601\pi$
$758$	−8.37682	−0.304260
$759$	0.124053	0.00450283
$760$	13.8892	0.503813
$761$	30.4564	1.10404	0.552022	−	0.833830i	$-0.313856\pi$
$761$	30.4564	1.10404	0.552022	+	0.833830i	$0.313856\pi$
$762$	3.48242	0.126155
$763$	−6.03454	−0.218465
$764$	16.8557	0.609819
$765$	10.6430	0.384798
$766$	2.84574	0.102821
$767$	−14.3798	−0.519225
$768$	9.13362	0.329581
$769$	10.5829	0.381630	0.190815	−	0.981626i	$-0.438887\pi$
$769$	10.5829	0.381630	0.190815	+	0.981626i	$0.438887\pi$
$770$	−0.150929	−0.00543911
$771$	9.04017	0.325574
$772$	44.9068	1.61623
$773$	−27.8870	−1.00302	−0.501512	−	0.865151i	$-0.667223\pi$
$773$	−27.8870	−1.00302	−0.501512	+	0.865151i	$0.667223\pi$
$774$	−4.82892	−0.173572
$775$	−1.36889	−0.0491721
$776$	10.8173	0.388318
$777$	−1.28533	−0.0461111
$778$	−8.52107	−0.305495
$779$	74.7251	2.67730
$780$	−3.54134	−0.126800
$781$	3.06764	0.109769
$782$	−0.0977636	−0.00349602
$783$	−20.5394	−0.734018
$784$	−24.7792	−0.884972
$785$	10.4286	0.372212
$786$	0.223289	0.00796444
$787$	−19.4332	−0.692718	−0.346359	−	0.938102i	$-0.612582\pi$
$787$	−19.4332	−0.692718	−0.346359	+	0.938102i	$0.612582\pi$
$788$	−50.6953	−1.80595
$789$	11.3274	0.403266
$790$	−5.02019	−0.178610
$791$	−1.32282	−0.0470342
$792$	−2.35459	−0.0836669
$793$	8.53182	0.302974
$794$	−7.53579	−0.267435
$795$	−7.69369	−0.272867
$796$	−17.4871	−0.619815
$797$	19.6266	0.695209	0.347604	−	0.937641i	$-0.386995\pi$
$797$	19.6266	0.695209	0.347604	+	0.937641i	$0.386995\pi$
$798$	−0.512036	−0.0181259
$799$	−6.11527	−0.216343
$800$	−4.92202	−0.174020
$801$	−20.8861	−0.737973
$802$	−1.67275	−0.0590668
$803$	−3.48730	−0.123064
$804$	18.7626	0.661705
$805$	−0.0838519	−0.00295539
$806$	−0.259087	−0.00912596
$807$	−18.2402	−0.642087
$808$	−5.04569	−0.177507
$809$	−43.0027	−1.51189	−0.755947	−	0.654633i	$-0.772823\pi$
$809$	−43.0027	−1.51189	−0.755947	+	0.654633i	$0.772823\pi$
$810$	−1.45997	−0.0512982
$811$	−25.6408	−0.900369	−0.450184	−	0.892936i	$-0.648642\pi$
$811$	−25.6408	−0.900369	−0.450184	+	0.892936i	$0.648642\pi$
$812$	2.76847	0.0971543
$813$	−22.3949	−0.785424
$814$	−1.27107	−0.0445509
$815$	28.0908	0.983980
$816$	7.76458	0.271815
$817$	59.0675	2.06651
$818$	−8.32550	−0.291094
$819$	−0.839425	−0.0293319
$820$	36.1242	1.26151
$821$	33.9321	1.18424	0.592119	−	0.805850i	$-0.298291\pi$
$821$	33.9321	1.18424	0.592119	+	0.805850i	$0.298291\pi$
$822$	−3.59555	−0.125409
$823$	−13.1596	−0.458713	−0.229357	−	0.973342i	$-0.573662\pi$
$823$	−13.1596	−0.458713	−0.229357	+	0.973342i	$0.573662\pi$
$824$	−6.77361	−0.235970
$825$	1.38868	0.0483475
$826$	−1.00512	−0.0349726
$827$	37.1147	1.29060	0.645302	−	0.763928i	$-0.276732\pi$
$827$	37.1147	1.29060	0.645302	+	0.763928i	$0.276732\pi$
$828$	−0.642573	−0.0223310
$829$	−11.3361	−0.393718	−0.196859	−	0.980432i	$-0.563074\pi$
$829$	−11.3361	−0.393718	−0.196859	+	0.980432i	$0.563074\pi$
$830$	2.18736	0.0759243
$831$	−16.7316	−0.580413
$832$	7.52026	0.260718
$833$	−17.5746	−0.608923
$834$	−2.21879	−0.0768303
$835$	−21.8524	−0.756233
$836$	14.1475	0.489303
$837$	−3.75393	−0.129755
$838$	4.36762	0.150877
$839$	8.72748	0.301306	0.150653	−	0.988587i	$-0.451862\pi$
$839$	8.72748	0.301306	0.150653	+	0.988587i	$0.451862\pi$
$840$	−0.503924	−0.0173870
$841$	−8.01336	−0.276323
$842$	3.27407	0.112832
$843$	1.07818	0.0371345
$844$	−31.7238	−1.09198
$845$	21.3057	0.732938
$846$	1.43858	0.0494594
$847$	−0.312976	−0.0107540
$848$	17.7274	0.608763
$849$	26.3646	0.904832
$850$	−1.09439	−0.0375373
$851$	−0.706169	−0.0242072
$852$	5.03111	0.172363
$853$	−32.0537	−1.09750	−0.548749	−	0.835987i	$-0.684896\pi$
$853$	−32.0537	−1.09750	−0.548749	+	0.835987i	$0.684896\pi$
$854$	0.596357	0.0204069
$855$	30.6252	1.04736
$856$	14.4191	0.492836
$857$	−44.6572	−1.52546	−0.762730	−	0.646717i	$-0.776141\pi$
$857$	−44.6572	−1.52546	−0.762730	+	0.646717i	$0.776141\pi$
$858$	0.262832	0.00897293
$859$	−27.5163	−0.938845	−0.469423	−	0.882974i	$-0.655538\pi$
$859$	−27.5163	−0.938845	−0.469423	+	0.882974i	$0.655538\pi$
$860$	28.5549	0.973713
$861$	−2.71116	−0.0923962
$862$	0.755678	0.0257385
$863$	14.5849	0.496475	0.248238	−	0.968699i	$-0.420149\pi$
$863$	14.5849	0.496475	0.248238	+	0.968699i	$0.420149\pi$
$864$	−13.4977	−0.459201
$865$	−41.2489	−1.40250
$866$	10.8336	0.368140
$867$	−8.93244	−0.303361
$868$	0.505985	0.0171743
$869$	−10.4102	−0.353141
$870$	−1.87644	−0.0636173
$871$	13.4662	0.456284
$872$	−19.9246	−0.674732
$873$	23.8518	0.807261
$874$	−0.281315	−0.00951563
$875$	−3.80929	−0.128778
$876$	−5.71937	−0.193239
$877$	20.8976	0.705662	0.352831	−	0.935687i	$-0.385219\pi$
$877$	20.8976	0.705662	0.352831	+	0.935687i	$0.385219\pi$
$878$	−1.98970	−0.0671491
$879$	−13.2499	−0.446908
$880$	6.58577	0.222006
$881$	43.3494	1.46048	0.730238	−	0.683192i	$-0.239409\pi$
$881$	43.3494	1.46048	0.730238	+	0.683192i	$0.239409\pi$
$882$	4.13431	0.139209
$883$	11.4401	0.384990	0.192495	−	0.981298i	$-0.438342\pi$
$883$	11.4401	0.384990	0.192495	+	0.981298i	$0.438342\pi$
$884$	5.78730	0.194648
$885$	−19.0344	−0.639835
$886$	5.98494	0.201068
$887$	−26.3967	−0.886315	−0.443157	−	0.896444i	$-0.646142\pi$
$887$	−26.3967	−0.886315	−0.443157	+	0.896444i	$0.646142\pi$
$888$	−4.24386	−0.142415
$889$	−4.88119	−0.163710
$890$	−4.42038	−0.148171
$891$	−3.02749	−0.101425
$892$	15.1659	0.507792
$893$	−17.5967	−0.588852
$894$	0.645007	0.0215723
$895$	−5.22195	−0.174551
$896$	2.41010	0.0805157
$897$	0.146022	0.00487553
$898$	9.66725	0.322600
$899$	3.83566	0.127927
$900$	−7.19313	−0.239771
$901$	12.5731	0.418872
$902$	−2.68108	−0.0892700
$903$	−2.14308	−0.0713171
$904$	−4.36764	−0.145266
$905$	−31.7019	−1.05381
$906$	−0.613606	−0.0203857
$907$	−2.78495	−0.0924728	−0.0462364	−	0.998931i	$-0.514723\pi$
$907$	−2.78495	−0.0924728	−0.0462364	+	0.998931i	$0.514723\pi$
$908$	14.2072	0.471482
$909$	−11.1256	−0.369013
$910$	−0.177658	−0.00588930
$911$	27.7962	0.920931	0.460465	−	0.887678i	$-0.347683\pi$
$911$	27.7962	0.920931	0.460465	+	0.887678i	$0.347683\pi$
$912$	22.3426	0.739838
$913$	4.53584	0.150114
$914$	−7.00853	−0.231822
$915$	11.2935	0.373351
$916$	−45.6313	−1.50770
$917$	−0.312976	−0.0103354
$918$	−3.00116	−0.0990529
$919$	54.6357	1.80227	0.901133	−	0.433543i	$-0.142737\pi$
$919$	54.6357	1.80227	0.901133	+	0.433543i	$0.142737\pi$
$920$	−0.276859	−0.00912776
$921$	−2.55141	−0.0840718
$922$	−4.10159	−0.135079
$923$	3.61090	0.118854
$924$	−0.513299	−0.0168863
$925$	−7.90504	−0.259916
$926$	3.60177	0.118361
$927$	−14.9356	−0.490550
$928$	13.7916	0.452731
$929$	−20.7158	−0.679663	−0.339832	−	0.940486i	$-0.610370\pi$
$929$	−20.7158	−0.679663	−0.339832	+	0.940486i	$0.610370\pi$
$930$	−0.342951	−0.0112458
$931$	−50.5709	−1.65739
$932$	−27.7155	−0.907850
$933$	−24.5614	−0.804103
$934$	−4.66247	−0.152561
$935$	4.67093	0.152756
$936$	−2.77158	−0.0905920
$937$	10.2275	0.334118	0.167059	−	0.985947i	$-0.446573\pi$
$937$	10.2275	0.334118	0.167059	+	0.985947i	$0.446573\pi$
$938$	0.941259	0.0307332
$939$	16.6368	0.542920
$940$	−8.50675	−0.277460
$941$	39.6026	1.29101	0.645504	−	0.763757i	$-0.276647\pi$
$941$	39.6026	1.29101	0.645504	+	0.763757i	$0.276647\pi$
$942$	−1.26939	−0.0413590
$943$	−1.48953	−0.0485057
$944$	43.8582	1.42746
$945$	−2.57409	−0.0837352
$946$	−2.11929	−0.0689041
$947$	36.5503	1.18773	0.593863	−	0.804566i	$-0.297602\pi$
$947$	36.5503	1.18773	0.593863	+	0.804566i	$0.297602\pi$
$948$	−17.0733	−0.554514
$949$	−4.10488	−0.133250
$950$	−3.14911	−0.102171
$951$	−14.5442	−0.471630
$952$	0.823520	0.0266904
$953$	−5.66479	−0.183501	−0.0917503	−	0.995782i	$-0.529246\pi$
$953$	−5.66479	−0.183501	−0.0917503	+	0.995782i	$0.529246\pi$
$954$	−2.95775	−0.0957607
$955$	16.0135	0.518185
$956$	14.3845	0.465230
$957$	−3.89110	−0.125781
$958$	−1.11551	−0.0360404
$959$	5.03975	0.162742
$960$	9.95450	0.321280
$961$	−30.2990	−0.977386
$962$	−1.49617	−0.0482384
$963$	31.7938	1.02454
$964$	−17.8583	−0.575177
$965$	42.6629	1.37337
$966$	0.0102066	0.000328393 0
$967$	5.75688	0.185129	0.0925644	−	0.995707i	$-0.470494\pi$
$967$	5.75688	0.185129	0.0925644	+	0.995707i	$0.470494\pi$
$968$	−1.03337	−0.0332138
$969$	15.8464	0.509060
$970$	5.04805	0.162083
$971$	0.266416	0.00854970	0.00427485	−	0.999991i	$-0.498639\pi$
$971$	0.266416	0.00854970	0.00427485	+	0.999991i	$0.498639\pi$
$972$	−30.9367	−0.992294
$973$	3.11000	0.0997019
$974$	−2.42838	−0.0778104
$975$	1.63460	0.0523493
$976$	−26.0220	−0.832942
$977$	−38.1250	−1.21973	−0.609863	−	0.792507i	$-0.708776\pi$
$977$	−38.1250	−1.21973	−0.609863	+	0.792507i	$0.708776\pi$
$978$	−3.41928	−0.109337
$979$	−9.16636	−0.292958
$980$	−24.4474	−0.780943
$981$	−43.9332	−1.40268
$982$	1.54112	0.0491791
$983$	22.0392	0.702941	0.351471	−	0.936199i	$-0.385682\pi$
$983$	22.0392	0.702941	0.351471	+	0.936199i	$0.385682\pi$
$984$	−8.95161	−0.285367
$985$	−48.1623	−1.53458
$986$	3.06650	0.0976574
$987$	0.638442	0.0203218
$988$	16.6530	0.529802
$989$	−1.17742	−0.0374397
$990$	−1.09881	−0.0349224
$991$	58.3980	1.85507	0.927536	−	0.373734i	$-0.121923\pi$
$991$	58.3980	1.85507	0.927536	+	0.373734i	$0.121923\pi$
$992$	2.52065	0.0800307
$993$	−9.04862	−0.287149
$994$	0.252395	0.00800548
$995$	−16.6134	−0.526679
$996$	7.43904	0.235715
$997$	−35.6413	−1.12877	−0.564385	−	0.825511i	$-0.690887\pi$
$997$	−35.6413	−1.12877	−0.564385	+	0.825511i	$0.690887\pi$
$998$	5.16886	0.163617
$999$	−21.6781	−0.685863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1441.2.a.f.1.16	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1441.2.a.f.1.16	✓	31	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:	\( N \)	\(=\)	\( 1441 = 11 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1441.a (trivial)

\(f(q)\)	\(=\)	\(q+0.262885 q^{2} +0.849379 q^{3} -1.93089 q^{4} -1.83441 q^{5} +0.223289 q^{6} -0.312976 q^{7} -1.03337 q^{8} -2.27856 q^{9} +O(q^{10})\)	\(q+0.262885 q^{2} +0.849379 q^{3} -1.93089 q^{4} -1.83441 q^{5} +0.223289 q^{6} -0.312976 q^{7} -1.03337 q^{8} -2.27856 q^{9} -0.482239 q^{10} -1.00000 q^{11} -1.64006 q^{12} -1.17709 q^{13} -0.0822766 q^{14} -1.55811 q^{15} +3.59013 q^{16} +2.54628 q^{17} -0.598997 q^{18} +7.32694 q^{19} +3.54205 q^{20} -0.265835 q^{21} -0.262885 q^{22} -0.146051 q^{23} -0.877723 q^{24} -1.63493 q^{25} -0.309440 q^{26} -4.48349 q^{27} +0.604323 q^{28} +4.58112 q^{29} -0.409603 q^{30} +0.837277 q^{31} +3.01053 q^{32} -0.849379 q^{33} +0.669379 q^{34} +0.574127 q^{35} +4.39964 q^{36} +4.83508 q^{37} +1.92614 q^{38} -0.999799 q^{39} +1.89563 q^{40} +10.1987 q^{41} -0.0698840 q^{42} +8.06168 q^{43} +1.93089 q^{44} +4.17981 q^{45} -0.0383946 q^{46} -2.40165 q^{47} +3.04938 q^{48} -6.90205 q^{49} -0.429799 q^{50} +2.16276 q^{51} +2.27284 q^{52} +4.93783 q^{53} -1.17864 q^{54} +1.83441 q^{55} +0.323420 q^{56} +6.22335 q^{57} +1.20431 q^{58} +12.2163 q^{59} +3.00854 q^{60} -7.24820 q^{61} +0.220107 q^{62} +0.713133 q^{63} -6.38883 q^{64} +2.15928 q^{65} -0.223289 q^{66} -11.4402 q^{67} -4.91660 q^{68} -0.124053 q^{69} +0.150929 q^{70} -3.06764 q^{71} +2.35459 q^{72} +3.48730 q^{73} +1.27107 q^{74} -1.38868 q^{75} -14.1475 q^{76} +0.312976 q^{77} -0.262832 q^{78} +10.4102 q^{79} -6.58577 q^{80} +3.02749 q^{81} +2.68108 q^{82} -4.53584 q^{83} +0.513299 q^{84} -4.67093 q^{85} +2.11929 q^{86} +3.89110 q^{87} +1.03337 q^{88} +9.16636 q^{89} +1.09881 q^{90} +0.368402 q^{91} +0.282009 q^{92} +0.711165 q^{93} -0.631356 q^{94} -13.4406 q^{95} +2.55708 q^{96} -10.4679 q^{97} -1.81444 q^{98} +2.27856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9}+O(q^{10}) \) 31 * q + 6 * q^2 + 4 * q^3 + 38 * q^4 + 8 * q^5 + 7 * q^6 + 4 * q^7 + 24 * q^8 + 45 * q^9	\( 31 q + 6 q^{2} + 4 q^{3} + 38 q^{4} + 8 q^{5} + 7 q^{6} + 4 q^{7} + 24 q^{8} + 45 q^{9} - 8 q^{10} - 31 q^{11} + 10 q^{12} - 8 q^{13} + 29 q^{14} + 36 q^{15} + 52 q^{16} - q^{17} + 33 q^{18} - 2 q^{19} + 22 q^{20} - 13 q^{21} - 6 q^{22} + 45 q^{23} + 16 q^{24} + 41 q^{25} + 24 q^{26} + 22 q^{27} + 17 q^{28} + 5 q^{29} + 29 q^{30} + 28 q^{31} + 69 q^{32} - 4 q^{33} + 14 q^{34} + 36 q^{35} + 63 q^{36} - 3 q^{37} + 4 q^{38} + 40 q^{39} - 48 q^{40} + 21 q^{41} - 9 q^{42} - 20 q^{43} - 38 q^{44} + 28 q^{45} - 24 q^{46} + 57 q^{47} - 46 q^{48} + 37 q^{49} + 64 q^{50} + 17 q^{51} - 11 q^{52} + 32 q^{53} - 26 q^{54} - 8 q^{55} + 84 q^{56} + 10 q^{57} - 17 q^{58} + 70 q^{59} - 33 q^{60} - 51 q^{61} - 34 q^{62} + 32 q^{63} + 80 q^{64} - q^{65} - 7 q^{66} + 24 q^{67} - 13 q^{68} + 19 q^{69} - 9 q^{70} + 128 q^{71} + 118 q^{72} - 27 q^{73} - 23 q^{74} + 41 q^{75} - 34 q^{76} - 4 q^{77} + 9 q^{78} + 2 q^{79} - 45 q^{80} + 43 q^{81} - 18 q^{82} + 46 q^{83} - 103 q^{84} - 50 q^{85} + 78 q^{86} - 9 q^{87} - 24 q^{88} + 52 q^{89} - 46 q^{90} + 38 q^{91} + 54 q^{92} + 4 q^{93} + 3 q^{94} + 70 q^{95} - 21 q^{96} + 3 q^{97} - 120 q^{98} - 45 q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 + 4 * q^3 + 38 * q^4 + 8 * q^5 + 7 * q^6 + 4 * q^7 + 24 * q^8 + 45 * q^9 - 8 * q^10 - 31 * q^11 + 10 * q^12 - 8 * q^13 + 29 * q^14 + 36 * q^15 + 52 * q^16 - q^17 + 33 * q^18 - 2 * q^19 + 22 * q^20 - 13 * q^21 - 6 * q^22 + 45 * q^23 + 16 * q^24 + 41 * q^25 + 24 * q^26 + 22 * q^27 + 17 * q^28 + 5 * q^29 + 29 * q^30 + 28 * q^31 + 69 * q^32 - 4 * q^33 + 14 * q^34 + 36 * q^35 + 63 * q^36 - 3 * q^37 + 4 * q^38 + 40 * q^39 - 48 * q^40 + 21 * q^41 - 9 * q^42 - 20 * q^43 - 38 * q^44 + 28 * q^45 - 24 * q^46 + 57 * q^47 - 46 * q^48 + 37 * q^49 + 64 * q^50 + 17 * q^51 - 11 * q^52 + 32 * q^53 - 26 * q^54 - 8 * q^55 + 84 * q^56 + 10 * q^57 - 17 * q^58 + 70 * q^59 - 33 * q^60 - 51 * q^61 - 34 * q^62 + 32 * q^63 + 80 * q^64 - q^65 - 7 * q^66 + 24 * q^67 - 13 * q^68 + 19 * q^69 - 9 * q^70 + 128 * q^71 + 118 * q^72 - 27 * q^73 - 23 * q^74 + 41 * q^75 - 34 * q^76 - 4 * q^77 + 9 * q^78 + 2 * q^79 - 45 * q^80 + 43 * q^81 - 18 * q^82 + 46 * q^83 - 103 * q^84 - 50 * q^85 + 78 * q^86 - 9 * q^87 - 24 * q^88 + 52 * q^89 - 46 * q^90 + 38 * q^91 + 54 * q^92 + 4 * q^93 + 3 * q^94 + 70 * q^95 - 21 * q^96 + 3 * q^97 - 120 * q^98 - 45 * q^99

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