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Properties

Label 2320.2.g.b
Level $2320$
Weight $2$
Character orbit 2320.g
Analytic conductor $18.525$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2320,2,Mod(1681,2320)] 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("2320.1681"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,4] 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: no
Analytic conductor: \(18.5252932689\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 580)
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 \(\beta = 2\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + 2 q^{7} + 3 q^{9} - \beta q^{11} - 2 q^{13} - \beta q^{19} - 6 q^{23} + q^{25} + (\beta - 1) q^{29} - \beta q^{31} - 2 q^{35} - 2 \beta q^{43} - 3 q^{45} + 2 \beta q^{47} - 3 q^{49} - 2 q^{53} + \cdots - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{13} - 12 q^{23} + 2 q^{25} - 2 q^{29} - 4 q^{35} - 6 q^{45} - 6 q^{49} - 4 q^{53} + 12 q^{63} + 4 q^{65} - 20 q^{67} + 16 q^{71} + 18 q^{81} - 4 q^{83} - 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(1857\) \(2031\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1681.1
0.500000 + 1.32288i
0.500000 − 1.32288i
0 0 0 −1.00000 0 2.00000 0 3.00000 0
1681.2 0 0 0 −1.00000 0 2.00000 0 3.00000 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 2320.2.g.b 2
4.b odd 2 1 580.2.d.a ✓ 2
12.b even 2 1 5220.2.l.c 2
20.d odd 2 1 2900.2.d.b 2
20.e even 4 2 2900.2.f.a 4
29.b even 2 1 inner 2320.2.g.b 2
116.d odd 2 1 580.2.d.a ✓ 2
348.b even 2 1 5220.2.l.c 2
580.e odd 2 1 2900.2.d.b 2
580.o even 4 2 2900.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.d.a ✓ 2 4.b odd 2 1
580.2.d.a ✓ 2 116.d odd 2 1
2320.2.g.b 2 1.a even 1 1 trivial
2320.2.g.b 2 29.b even 2 1 inner
2900.2.d.b 2 20.d odd 2 1
2900.2.d.b 2 580.e odd 2 1
2900.2.f.a 4 20.e even 4 2
2900.2.f.a 4 580.o even 4 2
5220.2.l.c 2 12.b even 2 1
5220.2.l.c 2 348.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2320, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 28 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 28 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 29 \) Copy content Toggle raw display
$31$ \( T^{2} + 28 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 112 \) Copy content Toggle raw display
$47$ \( T^{2} + 112 \) Copy content Toggle raw display
$53$ \( (T + 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 112 \) Copy content Toggle raw display
$67$ \( (T + 10)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 112 \) Copy content Toggle raw display
$79$ \( T^{2} + 28 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 112 \) Copy content Toggle raw display
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