Properties

Label 3720.2.a.r.1.2
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $0$
Dimension $4$
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] = [3720,2,Mod(1,3720)] 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("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,-4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.09502\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.09502 q^{7} +1.00000 q^{9} +5.25711 q^{11} +1.24976 q^{13} +1.00000 q^{15} -2.64620 q^{17} +3.25711 q^{19} +1.09502 q^{21} -6.99097 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.89596 q^{29} +1.00000 q^{31} -5.25711 q^{33} +1.09502 q^{35} -3.24976 q^{37} -1.24976 q^{39} +4.00000 q^{41} -4.34477 q^{43} -1.00000 q^{45} +7.14571 q^{47} -5.80094 q^{49} +2.64620 q^{51} +6.80094 q^{53} -5.25711 q^{55} -3.25711 q^{57} -8.80829 q^{59} +2.91234 q^{61} -1.09502 q^{63} -1.24976 q^{65} +1.43979 q^{67} +6.99097 q^{69} -4.24073 q^{71} -8.69689 q^{73} -1.00000 q^{75} -5.75662 q^{77} +12.0933 q^{79} +1.00000 q^{81} +0.0433434 q^{83} +2.64620 q^{85} -5.89596 q^{87} +4.63885 q^{89} -1.36850 q^{91} -1.00000 q^{93} -3.25711 q^{95} +0.309478 q^{97} +5.25711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{5} + q^{7} + 4 q^{9} + q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} - 7 q^{19} - q^{21} - q^{23} + 4 q^{25} - 4 q^{27} + 2 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} + 2 q^{39}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.09502 −0.413877 −0.206939 − 0.978354i \(-0.566350\pi\)
−0.206939 + 0.978354i \(0.566350\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.25711 1.58508 0.792539 − 0.609822i \(-0.208759\pi\)
0.792539 + 0.609822i \(0.208759\pi\)
\(12\) 0 0
\(13\) 1.24976 0.346620 0.173310 − 0.984867i \(-0.444554\pi\)
0.173310 + 0.984867i \(0.444554\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.64620 −0.641798 −0.320899 − 0.947113i \(-0.603985\pi\)
−0.320899 + 0.947113i \(0.603985\pi\)
\(18\) 0 0
\(19\) 3.25711 0.747232 0.373616 − 0.927584i \(-0.378118\pi\)
0.373616 + 0.927584i \(0.378118\pi\)
\(20\) 0 0
\(21\) 1.09502 0.238952
\(22\) 0 0
\(23\) −6.99097 −1.45772 −0.728859 − 0.684664i \(-0.759949\pi\)
−0.728859 + 0.684664i \(0.759949\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.89596 1.09485 0.547426 − 0.836854i \(-0.315608\pi\)
0.547426 + 0.836854i \(0.315608\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) −5.25711 −0.915145
\(34\) 0 0
\(35\) 1.09502 0.185092
\(36\) 0 0
\(37\) −3.24976 −0.534257 −0.267128 − 0.963661i \(-0.586075\pi\)
−0.267128 + 0.963661i \(0.586075\pi\)
\(38\) 0 0
\(39\) −1.24976 −0.200121
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −4.34477 −0.662571 −0.331286 − 0.943530i \(-0.607482\pi\)
−0.331286 + 0.943530i \(0.607482\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 7.14571 1.04231 0.521155 − 0.853462i \(-0.325502\pi\)
0.521155 + 0.853462i \(0.325502\pi\)
\(48\) 0 0
\(49\) −5.80094 −0.828706
\(50\) 0 0
\(51\) 2.64620 0.370542
\(52\) 0 0
\(53\) 6.80094 0.934181 0.467090 − 0.884210i \(-0.345302\pi\)
0.467090 + 0.884210i \(0.345302\pi\)
\(54\) 0 0
\(55\) −5.25711 −0.708868
\(56\) 0 0
\(57\) −3.25711 −0.431414
\(58\) 0 0
\(59\) −8.80829 −1.14674 −0.573371 − 0.819296i \(-0.694365\pi\)
−0.573371 + 0.819296i \(0.694365\pi\)
\(60\) 0 0
\(61\) 2.91234 0.372886 0.186443 − 0.982466i \(-0.440304\pi\)
0.186443 + 0.982466i \(0.440304\pi\)
\(62\) 0 0
\(63\) −1.09502 −0.137959
\(64\) 0 0
\(65\) −1.24976 −0.155013
\(66\) 0 0
\(67\) 1.43979 0.175898 0.0879491 − 0.996125i \(-0.471969\pi\)
0.0879491 + 0.996125i \(0.471969\pi\)
\(68\) 0 0
\(69\) 6.99097 0.841614
\(70\) 0 0
\(71\) −4.24073 −0.503282 −0.251641 − 0.967821i \(-0.580970\pi\)
−0.251641 + 0.967821i \(0.580970\pi\)
\(72\) 0 0
\(73\) −8.69689 −1.01789 −0.508947 − 0.860798i \(-0.669965\pi\)
−0.508947 + 0.860798i \(0.669965\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −5.75662 −0.656027
\(78\) 0 0
\(79\) 12.0933 1.36061 0.680304 − 0.732931i \(-0.261848\pi\)
0.680304 + 0.732931i \(0.261848\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.0433434 0.00475755 0.00237878 − 0.999997i \(-0.499243\pi\)
0.00237878 + 0.999997i \(0.499243\pi\)
\(84\) 0 0
\(85\) 2.64620 0.287021
\(86\) 0 0
\(87\) −5.89596 −0.632113
\(88\) 0 0
\(89\) 4.63885 0.491717 0.245858 − 0.969306i \(-0.420930\pi\)
0.245858 + 0.969306i \(0.420930\pi\)
\(90\) 0 0
\(91\) −1.36850 −0.143458
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −3.25711 −0.334172
\(96\) 0 0
\(97\) 0.309478 0.0314228 0.0157114 − 0.999877i \(-0.494999\pi\)
0.0157114 + 0.999877i \(0.494999\pi\)
\(98\) 0 0
\(99\) 5.25711 0.528359
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 3720.2.a.r.1.2 ✓ 4
4.3 odd 2 7440.2.a.bx.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.r.1.2 ✓ 4 1.1 even 1 trivial
7440.2.a.bx.1.3 4 4.3 odd 2