Properties

Label 2960.1.fx.a
Level $2960$
Weight $1$
Character orbit 2960.fx
Analytic conductor $1.477$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
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(719,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.719");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.fx (of order \(18\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{4} q^{5} + \zeta_{18}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{18}^{4} q^{5} + \zeta_{18}^{7} q^{9} + (\zeta_{18}^{8} + \zeta_{18}^{5}) q^{13} + ( - \zeta_{18}^{3} - \zeta_{18}^{2}) q^{17} + \zeta_{18}^{8} q^{25} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{29} - \zeta_{18}^{2} q^{37} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{41} + \zeta_{18}^{2} q^{45} + \zeta_{18} q^{49} + (\zeta_{18}^{7} - \zeta_{18}) q^{53} + ( - \zeta_{18}^{4} - 1) q^{61} + (\zeta_{18}^{3} + 1) q^{65} + (\zeta_{18}^{6} + \zeta_{18}^{3}) q^{73} - \zeta_{18}^{5} q^{81} + (\zeta_{18}^{7} + \zeta_{18}^{6}) q^{85} + (\zeta_{18}^{8} + 1) q^{89} + ( - \zeta_{18}^{8} + \zeta_{18}^{4}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{17} + 3 q^{41} - 6 q^{61} + 9 q^{65} - 3 q^{85} + 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{18}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0 0 0 −0.173648 0.984808i 0 0 0 −0.766044 + 0.642788i 0
959.1 0 0 0 0.939693 0.342020i 0 0 0 −0.173648 + 0.984808i 0
1119.1 0 0 0 −0.766044 + 0.642788i 0 0 0 0.939693 + 0.342020i 0
1439.1 0 0 0 −0.766044 0.642788i 0 0 0 0.939693 0.342020i 0
2079.1 0 0 0 −0.173648 + 0.984808i 0 0 0 −0.766044 0.642788i 0
2639.1 0 0 0 0.939693 + 0.342020i 0 0 0 −0.173648 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.1
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.x even 18 1 inner
740.bs odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.fx.a 6
4.b odd 2 1 CM 2960.1.fx.a 6
5.b even 2 1 2960.1.fx.b yes 6
20.d odd 2 1 2960.1.fx.b yes 6
37.f even 9 1 2960.1.fx.b yes 6
148.p odd 18 1 2960.1.fx.b yes 6
185.x even 18 1 inner 2960.1.fx.a 6
740.bs odd 18 1 inner 2960.1.fx.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.fx.a 6 1.a even 1 1 trivial
2960.1.fx.a 6 4.b odd 2 1 CM
2960.1.fx.a 6 185.x even 18 1 inner
2960.1.fx.a 6 740.bs odd 18 1 inner
2960.1.fx.b yes 6 5.b even 2 1
2960.1.fx.b yes 6 20.d odd 2 1
2960.1.fx.b yes 6 37.f even 9 1
2960.1.fx.b yes 6 148.p odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} - 9T_{13}^{3} + 27 \) acting on \(S_{1}^{\mathrm{new}}(2960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$41$ \( T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} + 9T^{3} + 27 \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( (T^{2} + 3)^{3} \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{4} + 9 T^{2} - 9 T + 3 \) Copy content Toggle raw display
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