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Properties

Label 1520.2.a.m
Level $1520$
Weight $2$
Character orbit 1520.a
Self dual yes
Analytic conductor $12.137$
Analytic rank $0$
Dimension $2$
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] = [1520,2,Mod(1,1520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0] 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{5} + 2 \beta q^{7} - q^{9} + (2 \beta - 2) q^{11} + (\beta + 2) q^{13} - \beta q^{15} + (2 \beta + 2) q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + q^{25} - 4 \beta q^{27} + ( - 2 \beta + 2) q^{29} + \cdots + ( - 2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{17} + 2 q^{19} + 8 q^{21} - 8 q^{23} + 2 q^{25} + 4 q^{29} + 8 q^{33} + 12 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{45} + 2 q^{49} + 8 q^{51} + 12 q^{53}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.41421 0 −1.00000 0 −2.82843 0 −1.00000 0
1.2 0 1.41421 0 −1.00000 0 2.82843 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(19\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.2.a.m 2
4.b odd 2 1 760.2.a.f ✓ 2
5.b even 2 1 7600.2.a.ba 2
8.b even 2 1 6080.2.a.bg 2
8.d odd 2 1 6080.2.a.bf 2
12.b even 2 1 6840.2.a.z 2
20.d odd 2 1 3800.2.a.n 2
20.e even 4 2 3800.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.f ✓ 2 4.b odd 2 1
1520.2.a.m 2 1.a even 1 1 trivial
3800.2.a.n 2 20.d odd 2 1
3800.2.d.i 4 20.e even 4 2
6080.2.a.bf 2 8.d odd 2 1
6080.2.a.bg 2 8.b even 2 1
6840.2.a.z 2 12.b even 2 1
7600.2.a.ba 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1520))\):

\( T_{3}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 72 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$59$ \( T^{2} - 8 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 16T + 62 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 112 \) Copy content Toggle raw display
$73$ \( T^{2} + 4T - 196 \) Copy content Toggle raw display
$79$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$83$ \( (T + 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
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