Properties

Label 238.2.a.f
Level $238$
Weight $2$
Character orbit 238.a
Self dual yes
Analytic conductor $1.900$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [238,2,Mod(1,238)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("238.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,2] 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: \(1.90043956811\)
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: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta + 1) q^{3} + q^{4} + ( - \beta + 1) q^{5} + ( - \beta - 1) q^{6} - q^{7} - q^{8} + (2 \beta + 3) q^{9} + (\beta - 1) q^{10} + (\beta + 3) q^{11} + (\beta + 1) q^{12} + ( - 2 \beta + 2) q^{13}+ \cdots + (9 \beta + 19) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} - 2 q^{10} + 6 q^{11} + 2 q^{12} + 4 q^{13} + 2 q^{14} - 8 q^{15} + 2 q^{16} + 2 q^{17} - 6 q^{18} - 8 q^{19} + 2 q^{20}+ \cdots + 38 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−1.00000 −1.23607 1.00000 3.23607 1.23607 −1.00000 −1.00000 −1.47214 −3.23607
1.2 −1.00000 3.23607 1.00000 −1.23607 −3.23607 −1.00000 −1.00000 7.47214 1.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(17\) \( -1 \)

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 238.2.a.f 2
3.b odd 2 1 2142.2.a.x 2
4.b odd 2 1 1904.2.a.f 2
5.b even 2 1 5950.2.a.x 2
7.b odd 2 1 1666.2.a.o 2
8.b even 2 1 7616.2.a.n 2
8.d odd 2 1 7616.2.a.y 2
17.b even 2 1 4046.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.a.f 2 1.a even 1 1 trivial
1666.2.a.o 2 7.b odd 2 1
1904.2.a.f 2 4.b odd 2 1
2142.2.a.x 2 3.b odd 2 1
4046.2.a.v 2 17.b even 2 1
5950.2.a.x 2 5.b even 2 1
7616.2.a.n 2 8.b even 2 1
7616.2.a.y 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(238))\):

\( T_{3}^{2} - 2T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 180 \) Copy content Toggle raw display
$79$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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