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Properties

Label 131.1.b.a
Level $131$
Weight $1$
Character orbit 131.b
Self dual yes
Analytic conductor $0.065$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -131
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [131,1,Mod(130,131)] 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("131.130"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(131, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 131 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 131.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.0653775166549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.17161.1
Artin image: $D_5$
Artin field: Galois closure of 5.1.17161.1

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{3} + q^{4} - \beta q^{5} - \beta q^{7} + ( - \beta + 1) q^{9} + (\beta - 1) q^{11} + (\beta - 1) q^{12} + (\beta - 1) q^{13} - q^{15} + q^{16} - \beta q^{20} - q^{21} + \beta q^{25} + \cdots + (\beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{4} - q^{5} - q^{7} + q^{9} - q^{11} - q^{12} - q^{13} - 2 q^{15} + 2 q^{16} - q^{20} - 2 q^{21} + q^{25} - 2 q^{27} - q^{28} + 3 q^{33} + 3 q^{35} + q^{36} + 3 q^{39} - q^{41}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/131\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
130.1
−0.618034
1.61803
0 −1.61803 1.00000 0.618034 0 0.618034 0 1.61803 0
130.2 0 0.618034 1.00000 −1.61803 0 −1.61803 0 −0.618034 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
131.b odd 2 1 CM by \(\Q(\sqrt{-131}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 131.1.b.a ✓ 2
3.b odd 2 1 1179.1.c.a 2
4.b odd 2 1 2096.1.h.b 2
5.b even 2 1 3275.1.c.d 2
5.c odd 4 2 3275.1.d.a 4
131.b odd 2 1 CM 131.1.b.a ✓ 2
393.d even 2 1 1179.1.c.a 2
524.b even 2 1 2096.1.h.b 2
655.d odd 2 1 3275.1.c.d 2
655.e even 4 2 3275.1.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
131.1.b.a ✓ 2 1.a even 1 1 trivial
131.1.b.a ✓ 2 131.b odd 2 1 CM
1179.1.c.a 2 3.b odd 2 1
1179.1.c.a 2 393.d even 2 1
2096.1.h.b 2 4.b odd 2 1
2096.1.h.b 2 524.b even 2 1
3275.1.c.d 2 5.b even 2 1
3275.1.c.d 2 655.d odd 2 1
3275.1.d.a 4 5.c odd 4 2
3275.1.d.a 4 655.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(131, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$13$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$61$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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