Properties

Label 2240.1.ca.b
Level $2240$
Weight $1$
Character orbit 2240.ca
Analytic conductor $1.118$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -40
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,1,Mod(929,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.929");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2240.ca (of order \(6\), degree \(2\), 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: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.134456000.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{5} + q^{7} + \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{5} + q^{7} + \zeta_{12}^{4} q^{9} + \zeta_{12}^{5} q^{11} + \zeta_{12}^{3} q^{13} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{19} + ( - \zeta_{12}^{2} - 1) q^{23} + \zeta_{12}^{2} q^{25} - \zeta_{12} q^{35} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{37} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{41} - \zeta_{12}^{5} q^{45} + \zeta_{12}^{4} q^{47} + q^{49} + (\zeta_{12}^{3} + \zeta_{12}) q^{53} + q^{55} + \zeta_{12}^{4} q^{63} - \zeta_{12}^{4} q^{65} + \zeta_{12}^{5} q^{77} - \zeta_{12}^{2} q^{81} + \zeta_{12}^{3} q^{91} + ( - \zeta_{12}^{4} + 1) q^{95} - \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 2 q^{9} - 6 q^{23} + 2 q^{25} - 2 q^{47} + 4 q^{49} + 4 q^{55} - 2 q^{63} + 2 q^{65} - 2 q^{81} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{12}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
929.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 −0.866025 0.500000i 0 1.00000 0 −0.500000 + 0.866025i 0
929.2 0 0 0 0.866025 + 0.500000i 0 1.00000 0 −0.500000 + 0.866025i 0
1249.1 0 0 0 −0.866025 + 0.500000i 0 1.00000 0 −0.500000 0.866025i 0
1249.2 0 0 0 0.866025 0.500000i 0 1.00000 0 −0.500000 0.866025i 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
8.b even 2 1 inner
20.d odd 2 1 inner
28.f even 6 1 inner
35.i odd 6 1 inner
56.m even 6 1 inner
280.bk odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.1.ca.b yes 4
4.b odd 2 1 2240.1.ca.a 4
5.b even 2 1 2240.1.ca.a 4
7.d odd 6 1 2240.1.ca.a 4
8.b even 2 1 inner 2240.1.ca.b yes 4
8.d odd 2 1 2240.1.ca.a 4
20.d odd 2 1 inner 2240.1.ca.b yes 4
28.f even 6 1 inner 2240.1.ca.b yes 4
35.i odd 6 1 inner 2240.1.ca.b yes 4
40.e odd 2 1 CM 2240.1.ca.b yes 4
40.f even 2 1 2240.1.ca.a 4
56.j odd 6 1 2240.1.ca.a 4
56.m even 6 1 inner 2240.1.ca.b yes 4
140.s even 6 1 2240.1.ca.a 4
280.ba even 6 1 2240.1.ca.a 4
280.bk odd 6 1 inner 2240.1.ca.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.1.ca.a 4 4.b odd 2 1
2240.1.ca.a 4 5.b even 2 1
2240.1.ca.a 4 7.d odd 6 1
2240.1.ca.a 4 8.d odd 2 1
2240.1.ca.a 4 40.f even 2 1
2240.1.ca.a 4 56.j odd 6 1
2240.1.ca.a 4 140.s even 6 1
2240.1.ca.a 4 280.ba even 6 1
2240.1.ca.b yes 4 1.a even 1 1 trivial
2240.1.ca.b yes 4 8.b even 2 1 inner
2240.1.ca.b yes 4 20.d odd 2 1 inner
2240.1.ca.b yes 4 28.f even 6 1 inner
2240.1.ca.b yes 4 35.i odd 6 1 inner
2240.1.ca.b yes 4 40.e odd 2 1 CM
2240.1.ca.b yes 4 56.m even 6 1 inner
2240.1.ca.b yes 4 280.bk odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{2} + 3T_{23} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2240, [\chi])\). 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
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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