Properties

Label 3040.1.cc.b
Level $3040$
Weight $1$
Character orbit 3040.cc
Analytic conductor $1.517$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -40
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,1,Mod(239,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3040.cc (of order \(6\), degree \(2\), not 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: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.14440.1
Artin image: $C_6\times S_3$
Artin 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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} - q^{7} + \zeta_{6}^{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{5} - q^{7} + \zeta_{6}^{2} q^{9} + q^{11} - 2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{19} - \zeta_{6}^{2} q^{23} + \zeta_{6}^{2} q^{25} - \zeta_{6} q^{35} + q^{37} + \zeta_{6} q^{41} - q^{45} + 2 \zeta_{6}^{2} q^{47} + \zeta_{6}^{2} q^{53} + \zeta_{6} q^{55} + 2 \zeta_{6} q^{59} - \zeta_{6}^{2} q^{63} + 2 q^{65} - q^{77} - \zeta_{6} q^{81} - \zeta_{6}^{2} q^{89} + 2 \zeta_{6}^{2} q^{91} + \zeta_{6}^{2} q^{95} + \zeta_{6}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} - 2 q^{7} - q^{9} + 2 q^{11} + 2 q^{13} + q^{19} + q^{23} - q^{25} - q^{35} + 2 q^{37} + q^{41} - 2 q^{45} - 2 q^{47} - q^{53} + q^{55} + 2 q^{59} + q^{63} + 4 q^{65} - 2 q^{77} - q^{81} + q^{89} - 2 q^{91} - q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(1217\) \(1921\) \(2661\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 0
1679.1 0 0 0 0.500000 + 0.866025i 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 \(\Q(\sqrt{-10}) \)
19.c even 3 1 inner
760.bm odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.1.cc.b 2
4.b odd 2 1 760.1.bm.b yes 2
5.b even 2 1 3040.1.cc.a 2
8.b even 2 1 760.1.bm.a 2
8.d odd 2 1 3040.1.cc.a 2
19.c even 3 1 inner 3040.1.cc.b 2
20.d odd 2 1 760.1.bm.a 2
20.e even 4 2 3800.1.bd.f 4
40.e odd 2 1 CM 3040.1.cc.b 2
40.f even 2 1 760.1.bm.b yes 2
40.i odd 4 2 3800.1.bd.f 4
76.g odd 6 1 760.1.bm.b yes 2
95.i even 6 1 3040.1.cc.a 2
152.k odd 6 1 3040.1.cc.a 2
152.p even 6 1 760.1.bm.a 2
380.p odd 6 1 760.1.bm.a 2
380.v even 12 2 3800.1.bd.f 4
760.z even 6 1 760.1.bm.b yes 2
760.bm odd 6 1 inner 3040.1.cc.b 2
760.br odd 12 2 3800.1.bd.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.1.bm.a 2 8.b even 2 1
760.1.bm.a 2 20.d odd 2 1
760.1.bm.a 2 152.p even 6 1
760.1.bm.a 2 380.p odd 6 1
760.1.bm.b yes 2 4.b odd 2 1
760.1.bm.b yes 2 40.f even 2 1
760.1.bm.b yes 2 76.g odd 6 1
760.1.bm.b yes 2 760.z even 6 1
3040.1.cc.a 2 5.b even 2 1
3040.1.cc.a 2 8.d odd 2 1
3040.1.cc.a 2 95.i even 6 1
3040.1.cc.a 2 152.k odd 6 1
3040.1.cc.b 2 1.a even 1 1 trivial
3040.1.cc.b 2 19.c even 3 1 inner
3040.1.cc.b 2 40.e odd 2 1 CM
3040.1.cc.b 2 760.bm odd 6 1 inner
3800.1.bd.f 4 20.e even 4 2
3800.1.bd.f 4 40.i odd 4 2
3800.1.bd.f 4 380.v even 12 2
3800.1.bd.f 4 760.br odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3040, [\chi])\). 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
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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