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Properties

Label 57.2.a.a.1.1
Level $57$
Weight $2$
Character 57.1
Self dual yes
Analytic conductor $0.455$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [57,2,Mod(1,57)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("57.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(57, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.455147291521\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 57.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} +1.00000 q^{9} +6.00000 q^{10} +1.00000 q^{11} -2.00000 q^{12} +2.00000 q^{13} +10.0000 q^{14} +3.00000 q^{15} -4.00000 q^{16} -1.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -6.00000 q^{20} +5.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} -10.0000 q^{28} -2.00000 q^{29} -6.00000 q^{30} -6.00000 q^{31} +8.00000 q^{32} -1.00000 q^{33} +2.00000 q^{34} +15.0000 q^{35} +2.00000 q^{36} +2.00000 q^{38} -2.00000 q^{39} -10.0000 q^{42} -1.00000 q^{43} +2.00000 q^{44} -3.00000 q^{45} +8.00000 q^{46} -9.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} -8.00000 q^{50} +1.00000 q^{51} +4.00000 q^{52} +10.0000 q^{53} +2.00000 q^{54} -3.00000 q^{55} +1.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} +6.00000 q^{60} -1.00000 q^{61} +12.0000 q^{62} -5.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} -30.0000 q^{70} -12.0000 q^{71} -11.0000 q^{73} -4.00000 q^{75} -2.00000 q^{76} -5.00000 q^{77} +4.00000 q^{78} +16.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} +12.0000 q^{83} +10.0000 q^{84} +3.00000 q^{85} +2.00000 q^{86} +2.00000 q^{87} -6.00000 q^{89} +6.00000 q^{90} -10.0000 q^{91} -8.00000 q^{92} +6.00000 q^{93} +18.0000 q^{94} +3.00000 q^{95} -8.00000 q^{96} -10.0000 q^{97} -36.0000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 2.00000 0.816497
\(7\) −5.00000 −1.88982 −0.944911 − 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 6.00000 1.89737
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −2.00000 −0.577350
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 10.0000 2.67261
\(15\) 3.00000 0.774597
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −2.00000 −0.471405
\(19\) −1.00000 −0.229416
\(20\) −6.00000 −1.34164
\(21\) 5.00000 1.09109
\(22\) −2.00000 −0.426401
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) −10.0000 −1.88982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −6.00000 −1.09545
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 8.00000 1.41421
\(33\) −1.00000 −0.174078
\(34\) 2.00000 0.342997
\(35\) 15.0000 2.53546
\(36\) 2.00000 0.333333
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 2.00000 0.324443
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −10.0000 −1.54303
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 −0.447214
\(46\) 8.00000 1.17954
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 4.00000 0.577350
\(49\) 18.0000 2.57143
\(50\) −8.00000 −1.13137
\(51\) 1.00000 0.140028
\(52\) 4.00000 0.554700
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 2.00000 0.272166
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 4.00000 0.525226
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 6.00000 0.774597
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 12.0000 1.52400
\(63\) −5.00000 −0.629941
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) −30.0000 −3.58569
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) −2.00000 −0.229416
\(77\) −5.00000 −0.569803
\(78\) 4.00000 0.452911
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 12.0000 1.34164
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 10.0000 1.09109
\(85\) 3.00000 0.325396
\(86\) 2.00000 0.215666
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 6.00000 0.632456
\(91\) −10.0000 −1.04828
\(92\) −8.00000 −0.834058
\(93\) 6.00000 0.622171
\(94\) 18.0000 1.85656
\(95\) 3.00000 0.307794
\(96\) −8.00000 −0.816497
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −36.0000 −3.63655
\(99\) 1.00000 0.100504
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 57.2.a.a.1.1 ✓ 1
3.2 odd 2 171.2.a.d.1.1 1
4.3 odd 2 912.2.a.g.1.1 1
5.2 odd 4 1425.2.c.b.799.1 2
5.3 odd 4 1425.2.c.b.799.2 2
5.4 even 2 1425.2.a.j.1.1 1
7.6 odd 2 2793.2.a.b.1.1 1
8.3 odd 2 3648.2.a.r.1.1 1
8.5 even 2 3648.2.a.bh.1.1 1
11.10 odd 2 6897.2.a.f.1.1 1
12.11 even 2 2736.2.a.v.1.1 1
13.12 even 2 9633.2.a.o.1.1 1
15.14 odd 2 4275.2.a.b.1.1 1
19.18 odd 2 1083.2.a.e.1.1 1
21.20 even 2 8379.2.a.p.1.1 1
57.56 even 2 3249.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
171.2.a.d.1.1 1 3.2 odd 2
912.2.a.g.1.1 1 4.3 odd 2
1083.2.a.e.1.1 1 19.18 odd 2
1425.2.a.j.1.1 1 5.4 even 2
1425.2.c.b.799.1 2 5.2 odd 4
1425.2.c.b.799.2 2 5.3 odd 4
2736.2.a.v.1.1 1 12.11 even 2
2793.2.a.b.1.1 1 7.6 odd 2
3249.2.a.b.1.1 1 57.56 even 2
3648.2.a.r.1.1 1 8.3 odd 2
3648.2.a.bh.1.1 1 8.5 even 2
4275.2.a.b.1.1 1 15.14 odd 2
6897.2.a.f.1.1 1 11.10 odd 2
8379.2.a.p.1.1 1 21.20 even 2
9633.2.a.o.1.1 1 13.12 even 2