Properties

Label 3780.1.bt.c
Level $3780$
Weight $1$
Character orbit 3780.bt
Analytic conductor $1.886$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3780,1,Mod(2449,3780)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3780, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3780.2449");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3780 = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3780.bt (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: \(1.88646574775\)
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 1260)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.11340.2
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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{7} - \zeta_{6} q^{11} + \zeta_{6}^{2} q^{13} - q^{17} - \zeta_{6} q^{25} - \zeta_{6} q^{29} - q^{35} - \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} - q^{55} + 2 \zeta_{6} q^{65} + q^{71} - q^{73} + \zeta_{6}^{2} q^{77} + \zeta_{6} q^{79} - \zeta_{6} q^{83} + 2 \zeta_{6}^{2} q^{85} + 2 q^{91} + \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} - q^{7} - q^{11} - 2 q^{13} - 4 q^{17} - q^{25} - q^{29} - 2 q^{35} - q^{47} - q^{49} - 2 q^{55} + 2 q^{65} + 2 q^{71} - 2 q^{73} - q^{77} + q^{79} - q^{83} - 2 q^{85} + 4 q^{91} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3780\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\) \(1891\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0
3709.1 0 0 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
9.c even 3 1 inner
315.bg odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3780.1.bt.c 2
3.b odd 2 1 1260.1.bt.b 2
5.b even 2 1 3780.1.bt.a 2
7.b odd 2 1 3780.1.bt.a 2
9.c even 3 1 inner 3780.1.bt.c 2
9.d odd 6 1 1260.1.bt.b 2
15.d odd 2 1 1260.1.bt.d yes 2
21.c even 2 1 1260.1.bt.d yes 2
35.c odd 2 1 CM 3780.1.bt.c 2
45.h odd 6 1 1260.1.bt.d yes 2
45.j even 6 1 3780.1.bt.a 2
63.l odd 6 1 3780.1.bt.a 2
63.o even 6 1 1260.1.bt.d yes 2
105.g even 2 1 1260.1.bt.b 2
315.z even 6 1 1260.1.bt.b 2
315.bg odd 6 1 inner 3780.1.bt.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.1.bt.b 2 3.b odd 2 1
1260.1.bt.b 2 9.d odd 6 1
1260.1.bt.b 2 105.g even 2 1
1260.1.bt.b 2 315.z even 6 1
1260.1.bt.d yes 2 15.d odd 2 1
1260.1.bt.d yes 2 21.c even 2 1
1260.1.bt.d yes 2 45.h odd 6 1
1260.1.bt.d yes 2 63.o even 6 1
3780.1.bt.a 2 5.b even 2 1
3780.1.bt.a 2 7.b odd 2 1
3780.1.bt.a 2 45.j even 6 1
3780.1.bt.a 2 63.l odd 6 1
3780.1.bt.c 2 1.a even 1 1 trivial
3780.1.bt.c 2 9.c even 3 1 inner
3780.1.bt.c 2 35.c odd 2 1 CM
3780.1.bt.c 2 315.bg odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3780, [\chi])\):

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