Properties

Label 2760.1.o.c
Level $2760$
Weight $1$
Character orbit 2760.o
Self dual yes
Analytic conductor $1.377$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -15, -2760, 184
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,1,Mod(1379,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.1379");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2760.o (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(1.37741943487\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-15}, \sqrt{46})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.41400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} + q^{15} + q^{16} + q^{18} - q^{20} + q^{23} - q^{24} + q^{25} - q^{27} + q^{30} + q^{32} + q^{36} - q^{40} - q^{45} + q^{46} - q^{48} + q^{49} + q^{50} + 2 q^{53} - q^{54} + q^{60} + 2 q^{61} + q^{64} - q^{69} + q^{72} - q^{75} - 2 q^{79} - q^{80} + q^{81} - q^{90} + q^{92} - q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1379.1
0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
184.h even 2 1 RM by \(\Q(\sqrt{46}) \)
2760.o odd 2 1 CM by \(\Q(\sqrt{-690}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.1.o.c yes 1
3.b odd 2 1 2760.1.o.b yes 1
5.b even 2 1 2760.1.o.b yes 1
8.d odd 2 1 2760.1.o.d yes 1
15.d odd 2 1 CM 2760.1.o.c yes 1
23.b odd 2 1 2760.1.o.d yes 1
24.f even 2 1 2760.1.o.a 1
40.e odd 2 1 2760.1.o.a 1
69.c even 2 1 2760.1.o.a 1
115.c odd 2 1 2760.1.o.a 1
120.m even 2 1 2760.1.o.d yes 1
184.h even 2 1 RM 2760.1.o.c yes 1
345.h even 2 1 2760.1.o.d yes 1
552.h odd 2 1 2760.1.o.b yes 1
920.b even 2 1 2760.1.o.b yes 1
2760.o odd 2 1 CM 2760.1.o.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.o.a 1 24.f even 2 1
2760.1.o.a 1 40.e odd 2 1
2760.1.o.a 1 69.c even 2 1
2760.1.o.a 1 115.c odd 2 1
2760.1.o.b yes 1 3.b odd 2 1
2760.1.o.b yes 1 5.b even 2 1
2760.1.o.b yes 1 552.h odd 2 1
2760.1.o.b yes 1 920.b even 2 1
2760.1.o.c yes 1 1.a even 1 1 trivial
2760.1.o.c yes 1 15.d odd 2 1 CM
2760.1.o.c yes 1 184.h even 2 1 RM
2760.1.o.c yes 1 2760.o odd 2 1 CM
2760.1.o.d yes 1 8.d odd 2 1
2760.1.o.d yes 1 23.b odd 2 1
2760.1.o.d yes 1 120.m even 2 1
2760.1.o.d yes 1 345.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2760, [\chi])\):

\( T_{53} - 2 \) Copy content Toggle raw display
\( T_{61} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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