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Properties

Label 4680.2.g.h
Level $4680$
Weight $2$
Character orbit 4680.g
Analytic conductor $37.370$
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] = [4680,2,Mod(2521,4680)] 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("4680.2521"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.3699881460\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (\beta_{2} + \beta_1) q^{11} + (3 \beta_{2} + 2) q^{13} + (\beta_{3} - 4) q^{17} - 2 \beta_1 q^{19} + \beta_{3} q^{23} - q^{25} - 2 q^{29} + ( - 2 \beta_{2} + 2 \beta_1) q^{31}+ \cdots + (5 \beta_{2} - 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{13} - 14 q^{17} + 2 q^{23} - 4 q^{25} - 8 q^{29} + 6 q^{35} + 20 q^{43} - 22 q^{49} - 22 q^{53} - 2 q^{55} + 14 q^{61} - 12 q^{65} - 38 q^{77} - 26 q^{79} + 18 q^{91} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 11\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{2} - 11\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(2081\) \(2341\) \(3511\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2521.1
− 2.70156i
3.70156i
− 3.70156i
2.70156i
0 0 0 − 1.00000i 0 − 1.70156i 0 0 0
2521.2 0 0 0 − 1.00000i 0 4.70156i 0 0 0
2521.3 0 0 0 1.00000i 0 − 4.70156i 0 0 0
2521.4 0 0 0 1.00000i 0 1.70156i 0 0 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4680.2.g.h 4
3.b odd 2 1 1560.2.g.f ✓ 4
12.b even 2 1 3120.2.g.r 4
13.b even 2 1 inner 4680.2.g.h 4
39.d odd 2 1 1560.2.g.f ✓ 4
156.h even 2 1 3120.2.g.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.g.f ✓ 4 3.b odd 2 1
1560.2.g.f ✓ 4 39.d odd 2 1
3120.2.g.r 4 12.b even 2 1
3120.2.g.r 4 156.h even 2 1
4680.2.g.h 4 1.a even 1 1 trivial
4680.2.g.h 4 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4680, [\chi])\):

\( T_{7}^{4} + 25T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} + 21T_{11}^{2} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 25T^{2} + 64 \) Copy content Toggle raw display
$11$ \( T^{4} + 21T^{2} + 100 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 7 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 84T^{2} + 1600 \) Copy content Toggle raw display
$23$ \( (T^{2} - T - 10)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 100T^{2} + 1024 \) Copy content Toggle raw display
$37$ \( T^{4} + 21T^{2} + 100 \) Copy content Toggle raw display
$41$ \( T^{4} + 81T^{2} + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 11 T + 20)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 180T^{2} + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 2)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 33T^{2} + 16 \) Copy content Toggle raw display
$73$ \( (T^{2} + 164)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 13 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 100T^{2} + 1024 \) Copy content Toggle raw display
$89$ \( T^{4} + 25T^{2} + 64 \) Copy content Toggle raw display
$97$ \( T^{4} + 269T^{2} + 2500 \) Copy content Toggle raw display
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