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Properties

Label 7440.2.a.r.1.1
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
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] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,1,0,-1,0,0,0,1,0,6,0,-2,0,-1,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.4086991038\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7440.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +6.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} -4.00000 q^{17} -2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.00000 q^{29} -1.00000 q^{31} +6.00000 q^{33} -6.00000 q^{37} -2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} -4.00000 q^{47} -7.00000 q^{49} -4.00000 q^{51} -6.00000 q^{53} -6.00000 q^{55} +4.00000 q^{61} +2.00000 q^{65} +4.00000 q^{67} -2.00000 q^{69} +8.00000 q^{71} -4.00000 q^{73} +1.00000 q^{75} +4.00000 q^{79} +1.00000 q^{81} +4.00000 q^{85} -8.00000 q^{87} -2.00000 q^{89} -1.00000 q^{93} +14.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
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 7440.2.a.r.1.1 1
4.3 odd 2 930.2.a.l.1.1 ✓ 1
12.11 even 2 2790.2.a.j.1.1 1
20.3 even 4 4650.2.d.a.3349.1 2
20.7 even 4 4650.2.d.a.3349.2 2
20.19 odd 2 4650.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.l.1.1 ✓ 1 4.3 odd 2
2790.2.a.j.1.1 1 12.11 even 2
4650.2.a.r.1.1 1 20.19 odd 2
4650.2.d.a.3349.1 2 20.3 even 4
4650.2.d.a.3349.2 2 20.7 even 4
7440.2.a.r.1.1 1 1.1 even 1 trivial