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Properties

Label 59.4.a.b
Level $59$
Weight $4$
Character orbit 59.a
Self dual yes
Analytic conductor $3.481$
Analytic rank $1$
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] = [59,4,Mod(1,59)] 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("59.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(59, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 59.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.48111269034\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta + 1) q^{3} + (\beta - 4) q^{4} + (3 \beta - 17) q^{5} + ( - 2 \beta - 12) q^{6} + (2 \beta - 5) q^{7} + ( - 11 \beta + 4) q^{8} + (3 \beta + 10) q^{9} + ( - 14 \beta + 12) q^{10}+ \cdots + (31 \beta + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - 7 q^{4} - 31 q^{5} - 26 q^{6} - 8 q^{7} - 3 q^{8} + 23 q^{9} + 10 q^{10} + 13 q^{11} - 22 q^{12} - 95 q^{13} + 13 q^{14} - 61 q^{15} - 39 q^{16} + 169 q^{17} + 37 q^{18} - 107 q^{19}+ \cdots + 175 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 5.68466 −5.56155 −21.6847 −8.87689 −8.12311 21.1771 5.31534 33.8617
1.2 2.56155 −6.68466 −1.43845 −9.31534 −17.1231 0.123106 −24.1771 17.6847 −23.8617
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \( -1 \)

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 59.4.a.b ✓ 2
3.b odd 2 1 531.4.a.a 2
4.b odd 2 1 944.4.a.e 2
5.b even 2 1 1475.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.4.a.b ✓ 2 1.a even 1 1 trivial
531.4.a.a 2 3.b odd 2 1
944.4.a.e 2 4.b odd 2 1
1475.4.a.a 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} + T - 38 \) Copy content Toggle raw display
$5$ \( T^{2} + 31T + 202 \) Copy content Toggle raw display
$7$ \( T^{2} + 8T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$13$ \( T^{2} + 95T + 1912 \) Copy content Toggle raw display
$17$ \( T^{2} - 169T + 7102 \) Copy content Toggle raw display
$19$ \( T^{2} + 107T - 236 \) Copy content Toggle raw display
$23$ \( T^{2} + 96T - 144 \) Copy content Toggle raw display
$29$ \( T^{2} + 307T + 7748 \) Copy content Toggle raw display
$31$ \( T^{2} - 56T - 9008 \) Copy content Toggle raw display
$37$ \( T^{2} + 291T + 9232 \) Copy content Toggle raw display
$41$ \( T^{2} - 452T + 36779 \) Copy content Toggle raw display
$43$ \( T^{2} + 35T - 11632 \) Copy content Toggle raw display
$47$ \( T^{2} - 26T - 47584 \) Copy content Toggle raw display
$53$ \( T^{2} + 641T + 5816 \) Copy content Toggle raw display
$59$ \( (T - 59)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 604T + 30004 \) Copy content Toggle raw display
$67$ \( T^{2} + 1326 T + 322456 \) Copy content Toggle raw display
$71$ \( T^{2} - 289T - 805664 \) Copy content Toggle raw display
$73$ \( T^{2} - 48T - 252452 \) Copy content Toggle raw display
$79$ \( T^{2} + 224T + 9671 \) Copy content Toggle raw display
$83$ \( T^{2} - 1963 T + 939436 \) Copy content Toggle raw display
$89$ \( T^{2} - 688T - 420292 \) Copy content Toggle raw display
$97$ \( T^{2} - 356 T - 1887548 \) Copy content Toggle raw display
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