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Properties

Label 1904.2.a.f
Level $1904$
Weight $2$
Character orbit 1904.a
Self dual yes
Analytic conductor $15.204$
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] = [1904,2,Mod(1,1904)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1904, 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("1904.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.2035165449\)
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: no (minimal twist has level 238)
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.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} + ( - \beta + 1) q^{5} + q^{7} + (2 \beta + 3) q^{9} + ( - \beta - 3) q^{11} + ( - 2 \beta + 2) q^{13} + 4 q^{15} + q^{17} + (2 \beta + 4) q^{19} + ( - \beta - 1) q^{21} - 8 q^{23}+ \cdots + ( - 9 \beta - 19) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9} - 6 q^{11} + 4 q^{13} + 8 q^{15} + 2 q^{17} + 8 q^{19} - 2 q^{21} - 16 q^{23} + 2 q^{25} - 20 q^{27} + 2 q^{29} + 12 q^{31} + 16 q^{33} + 2 q^{35} - 2 q^{37}+ \cdots - 38 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −3.23607 0 −1.23607 0 1.00000 0 7.47214 0
1.2 0 1.23607 0 3.23607 0 1.00000 0 −1.47214 0
\(n\): 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 1904.2.a.f 2
4.b odd 2 1 238.2.a.f ✓ 2
8.b even 2 1 7616.2.a.y 2
8.d odd 2 1 7616.2.a.n 2
12.b even 2 1 2142.2.a.x 2
20.d odd 2 1 5950.2.a.x 2
28.d even 2 1 1666.2.a.o 2
68.d odd 2 1 4046.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.a.f ✓ 2 4.b odd 2 1
1666.2.a.o 2 28.d even 2 1
1904.2.a.f 2 1.a even 1 1 trivial
2142.2.a.x 2 12.b even 2 1
4046.2.a.v 2 68.d odd 2 1
5950.2.a.x 2 20.d odd 2 1
7616.2.a.n 2 8.d odd 2 1
7616.2.a.y 2 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1904))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) 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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