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Properties

Label 580.2.d.b
Level $580$
Weight $2$
Character orbit 580.d
Analytic conductor $4.631$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [580,2,Mod(521,580)] 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("580.521"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(580, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 580 = 2^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 580.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-4] 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: no
Analytic conductor: \(4.63132331723\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.7168.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + (\beta_{3} + 2) q^{7} + (2 \beta_{3} - 3) q^{9} - \beta_{2} q^{11} + 4 \beta_{3} q^{13} - \beta_1 q^{15} + ( - \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{2} - \beta_1) q^{19}+ \cdots + (\beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 8 q^{7} - 12 q^{9} - 16 q^{23} + 4 q^{25} + 4 q^{29} + 8 q^{33} - 8 q^{35} + 12 q^{45} - 4 q^{49} - 40 q^{51} + 16 q^{53} + 8 q^{57} + 8 q^{59} - 8 q^{63} - 32 q^{67} + 24 q^{71} - 4 q^{81}+ \cdots - 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/580\mathbb{Z}\right)^\times\).

\(n\) \(117\) \(291\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
2.10100i
− 1.25928i
1.25928i
− 2.10100i
0 − 2.97127i 0 −1.00000 0 0.585786 0 −5.82843 0
521.2 0 − 1.78089i 0 −1.00000 0 3.41421 0 −0.171573 0
521.3 0 1.78089i 0 −1.00000 0 3.41421 0 −0.171573 0
521.4 0 2.97127i 0 −1.00000 0 0.585786 0 −5.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 580.2.d.b ✓ 4
3.b odd 2 1 5220.2.l.g 4
4.b odd 2 1 2320.2.g.d 4
5.b even 2 1 2900.2.d.c 4
5.c odd 4 2 2900.2.f.d 8
29.b even 2 1 inner 580.2.d.b ✓ 4
87.d odd 2 1 5220.2.l.g 4
116.d odd 2 1 2320.2.g.d 4
145.d even 2 1 2900.2.d.c 4
145.h odd 4 2 2900.2.f.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.d.b ✓ 4 1.a even 1 1 trivial
580.2.d.b ✓ 4 29.b even 2 1 inner
2320.2.g.d 4 4.b odd 2 1
2320.2.g.d 4 116.d odd 2 1
2900.2.d.c 4 5.b even 2 1
2900.2.d.c 4 145.d even 2 1
2900.2.f.d 8 5.c odd 4 2
2900.2.f.d 8 145.h odd 4 2
5220.2.l.g 4 3.b odd 2 1
5220.2.l.g 4 87.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 12T_{3}^{2} + 28 \) acting on \(S_{2}^{\mathrm{new}}(580, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 28 \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 20T^{2} + 28 \) Copy content Toggle raw display
$13$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 52T^{2} + 28 \) Copy content Toggle raw display
$19$ \( T^{4} + 76T^{2} + 1372 \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$31$ \( T^{4} + 12T^{2} + 28 \) Copy content Toggle raw display
$37$ \( T^{4} + 20T^{2} + 28 \) Copy content Toggle raw display
$41$ \( T^{4} + 40T^{2} + 112 \) Copy content Toggle raw display
$43$ \( T^{4} + 12T^{2} + 28 \) Copy content Toggle raw display
$47$ \( T^{4} + 12T^{2} + 28 \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 40T^{2} + 112 \) Copy content Toggle raw display
$67$ \( (T^{2} + 16 T + 62)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 332 T^{2} + 26908 \) Copy content Toggle raw display
$79$ \( T^{4} + 244 T^{2} + 14812 \) Copy content Toggle raw display
$83$ \( (T^{2} - 20 T + 50)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 384 T^{2} + 28672 \) Copy content Toggle raw display
$97$ \( T^{4} + 332 T^{2} + 26908 \) Copy content Toggle raw display
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