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Properties

Label 4761.2.a.k
Level $4761$
Weight $2$
Character orbit 4761.a
Self dual yes
Analytic conductor $38.017$
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] = [4761,2,Mod(1,4761)] 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("4761.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4761, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4761 = 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4761.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,4,0,4,-6,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.0167764023\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 207)
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.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta + 1) q^{4} + (\beta + 2) q^{5} + ( - \beta + 2) q^{7} + (\beta - 3) q^{8} + \beta q^{10} + 2 \beta q^{11} + (3 \beta - 4) q^{14} + 3 q^{16} + ( - \beta + 6) q^{17} + ( - 3 \beta + 2) q^{19}+ \cdots + (3 \beta - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 4 q^{5} + 4 q^{7} - 6 q^{8} - 8 q^{14} + 6 q^{16} + 12 q^{17} + 4 q^{19} - 4 q^{20} + 8 q^{22} + 2 q^{25} + 12 q^{28} + 6 q^{32} - 16 q^{34} + 4 q^{35} + 4 q^{37} - 16 q^{38} - 8 q^{40}+ \cdots - 14 q^{98}+O(q^{100}) \) 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
−2.41421 0 3.82843 0.585786 0 3.41421 −4.41421 0 −1.41421
1.2 0.414214 0 −1.82843 3.41421 0 0.585786 −1.58579 0 1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(23\) \( -1 \)

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 4761.2.a.k 2
3.b odd 2 1 4761.2.a.z 2
23.b odd 2 1 207.2.a.b ✓ 2
69.c even 2 1 207.2.a.e yes 2
92.b even 2 1 3312.2.a.u 2
115.c odd 2 1 5175.2.a.bo 2
276.h odd 2 1 3312.2.a.be 2
345.h even 2 1 5175.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.2.a.b ✓ 2 23.b odd 2 1
207.2.a.e yes 2 69.c even 2 1
3312.2.a.u 2 92.b even 2 1
3312.2.a.be 2 276.h odd 2 1
4761.2.a.k 2 1.a even 1 1 trivial
4761.2.a.z 2 3.b odd 2 1
5175.2.a.bc 2 345.h even 2 1
5175.2.a.bo 2 115.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4761))\):

\( T_{2}^{2} + 2T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12T + 34 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 14 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 72 \) Copy content Toggle raw display
$31$ \( T^{2} - 72 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 46 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 20T + 98 \) Copy content Toggle raw display
$71$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 94 \) Copy content Toggle raw display
$83$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T - 14 \) Copy content Toggle raw display
$97$ \( (T - 10)^{2} \) Copy content Toggle raw display
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