Properties

Label 2960.1.eo.a
Level $2960$
Weight $1$
Character orbit 2960.eo
Analytic conductor $1.477$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2960,1,Mod(1007,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 3, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.1007");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.eo (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{3} q^{5} - \zeta_{12} q^{9} - \zeta_{12}^{2} q^{13} + ( - \zeta_{12}^{2} - 1) q^{17} - q^{25} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{29} - \zeta_{12}^{5} q^{37} + (\zeta_{12}^{4} - 1) q^{41} - \zeta_{12}^{4} q^{45} - \zeta_{12} q^{49} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{53} + (\zeta_{12} + 1) q^{61} - 2 \zeta_{12}^{5} q^{65} + (\zeta_{12}^{3} + 1) q^{73} + \zeta_{12}^{2} q^{81} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{85} + (\zeta_{12}^{3} - \zeta_{12}^{2}) q^{89} - \zeta_{12}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{13} - 6 q^{17} - 4 q^{25} + 2 q^{29} - 6 q^{41} + 2 q^{45} + 2 q^{53} + 4 q^{61} + 4 q^{73} + 2 q^{81} - 2 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{5}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1007.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 1.00000i 0 0 0 0.866025 0.500000i 0
1583.1 0 0 0 1.00000i 0 0 0 −0.866025 + 0.500000i 0
2687.1 0 0 0 1.00000i 0 0 0 −0.866025 0.500000i 0
2863.1 0 0 0 1.00000i 0 0 0 0.866025 + 0.500000i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.p even 12 1 inner
740.bj odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.eo.a 4
4.b odd 2 1 CM 2960.1.eo.a 4
5.c odd 4 1 2960.1.et.a yes 4
20.e even 4 1 2960.1.et.a yes 4
37.g odd 12 1 2960.1.et.a yes 4
148.l even 12 1 2960.1.et.a yes 4
185.p even 12 1 inner 2960.1.eo.a 4
740.bj odd 12 1 inner 2960.1.eo.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.eo.a 4 1.a even 1 1 trivial
2960.1.eo.a 4 4.b odd 2 1 CM
2960.1.eo.a 4 185.p even 12 1 inner
2960.1.eo.a 4 740.bj odd 12 1 inner
2960.1.et.a yes 4 5.c odd 4 1
2960.1.et.a yes 4 20.e even 4 1
2960.1.et.a yes 4 37.g odd 12 1
2960.1.et.a yes 4 148.l even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

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