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Properties

Label 578.4.a.e
Level $578$
Weight $4$
Character orbit 578.a
Self dual yes
Analytic conductor $34.103$
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] = [578,4,Mod(1,578)] 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("578.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(578, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 578 = 2 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 578.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,-7,8,1,14,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.1031039833\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - 3 \beta - 2) q^{3} + 4 q^{4} + ( - \beta + 1) q^{5} + (6 \beta + 4) q^{6} + ( - 13 \beta - 3) q^{7} - 8 q^{8} + (21 \beta + 4) q^{9} + (2 \beta - 2) q^{10} + ( - 14 \beta - 7) q^{11}+ \cdots + ( - 497 \beta - 910) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 7 q^{3} + 8 q^{4} + q^{5} + 14 q^{6} - 19 q^{7} - 16 q^{8} + 29 q^{9} - 2 q^{10} - 28 q^{11} - 28 q^{12} + 45 q^{13} + 38 q^{14} + 16 q^{15} + 32 q^{16} - 58 q^{18} - 99 q^{19} + 4 q^{20}+ \cdots - 2317 q^{99}+O(q^{100}) \) 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
2.30278
−1.30278
−2.00000 −8.90833 4.00000 −1.30278 17.8167 −32.9361 −8.00000 52.3583 2.60555
1.2 −2.00000 1.90833 4.00000 2.30278 −3.81665 13.9361 −8.00000 −23.3583 −4.60555
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 578.4.a.e ✓ 2
17.b even 2 1 578.4.a.g yes 2
17.c even 4 2 578.4.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
578.4.a.e ✓ 2 1.a even 1 1 trivial
578.4.a.g yes 2 17.b even 2 1
578.4.b.f 4 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(578))\):

\( T_{3}^{2} + 7T_{3} - 17 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 7T - 17 \) Copy content Toggle raw display
$5$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 19T - 459 \) Copy content Toggle raw display
$11$ \( T^{2} + 28T - 441 \) Copy content Toggle raw display
$13$ \( T^{2} - 45T - 1525 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 99T + 1901 \) Copy content Toggle raw display
$23$ \( T^{2} - 111T + 3051 \) Copy content Toggle raw display
$29$ \( T^{2} + 91T - 66261 \) Copy content Toggle raw display
$31$ \( T^{2} - 58T - 3371 \) Copy content Toggle raw display
$37$ \( T^{2} - 438T + 47324 \) Copy content Toggle raw display
$41$ \( T^{2} - 364T + 22932 \) Copy content Toggle raw display
$43$ \( T^{2} - 350T + 28077 \) Copy content Toggle raw display
$47$ \( T^{2} - 384T + 36747 \) Copy content Toggle raw display
$53$ \( T^{2} + 933T + 207063 \) Copy content Toggle raw display
$59$ \( T^{2} + 720T - 117972 \) Copy content Toggle raw display
$61$ \( T^{2} + 188T - 641 \) Copy content Toggle raw display
$67$ \( T^{2} + 22T - 567732 \) Copy content Toggle raw display
$71$ \( T^{2} - 1332 T + 413604 \) Copy content Toggle raw display
$73$ \( T^{2} + 1284 T + 322607 \) Copy content Toggle raw display
$79$ \( T^{2} + 324T - 626044 \) Copy content Toggle raw display
$83$ \( T^{2} - 7T - 1633701 \) Copy content Toggle raw display
$89$ \( T^{2} + 1272 T + 366588 \) Copy content Toggle raw display
$97$ \( T^{2} + 2631 T + 1601837 \) Copy content Toggle raw display
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