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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2480,1,Mod(273,2480)] 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("2480.273"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2480, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 9, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2480 = 2^{4} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2480.de (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1240)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.1922000.4

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{3} - \zeta_{12} q^{5} + \zeta_{12} q^{9} - \zeta_{12}^{4} q^{11} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{13} + ( - \zeta_{12}^{3} - 1) q^{15} + ( - \zeta_{12}^{3} + 1) q^{23} + \cdots - \zeta_{12}^{5} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 4 q^{23} + 2 q^{25} + 2 q^{31} + 4 q^{33} - 2 q^{41} - 2 q^{45} - 2 q^{53} - 4 q^{61} + 2 q^{65} + 2 q^{67} - 2 q^{71} - 2 q^{75} - 2 q^{81} + 2 q^{87} + 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2480\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(561\) \(1551\) \(1861\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{4}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
273.1
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
0 −0.366025 + 1.36603i 0 0.866025 + 0.500000i 0 0 0 −0.866025 − 0.500000i 0
1617.1 0 −0.366025 − 1.36603i 0 0.866025 − 0.500000i 0 0 0 −0.866025 + 0.500000i 0
2113.1 0 1.36603 − 0.366025i 0 −0.866025 + 0.500000i 0 0 0 0.866025 − 0.500000i 0
2257.1 0 1.36603 + 0.366025i 0 −0.866025 − 0.500000i 0 0 0 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.c even 3 1 inner
155.o odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2480.1.de.b 4
4.b odd 2 1 1240.1.cg.a ✓ 4
5.c odd 4 1 inner 2480.1.de.b 4
20.e even 4 1 1240.1.cg.a ✓ 4
31.c even 3 1 inner 2480.1.de.b 4
124.i odd 6 1 1240.1.cg.a ✓ 4
155.o odd 12 1 inner 2480.1.de.b 4
620.be even 12 1 1240.1.cg.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1240.1.cg.a ✓ 4 4.b odd 2 1
1240.1.cg.a ✓ 4 20.e even 4 1
1240.1.cg.a ✓ 4 124.i odd 6 1
1240.1.cg.a ✓ 4 620.be even 12 1
2480.1.de.b 4 1.a even 1 1 trivial
2480.1.de.b 4 5.c odd 4 1 inner
2480.1.de.b 4 31.c even 3 1 inner
2480.1.de.b 4 155.o odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$59$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$61$ \( (T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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