Properties

Label 2775.1.x.a
Level $2775$
Weight $1$
Character orbit 2775.x
Analytic conductor $1.385$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2775,1,Mod(824,2775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2775, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("2775.824");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2775.x (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: \(1.38490541006\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.4107.1
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} - \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{7} - \zeta_{12}^{4} q^{9} - \zeta_{12} q^{12} - \zeta_{12}^{5} q^{13} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{2} q^{19} - \zeta_{12}^{4} q^{21} + \zeta_{12}^{3} q^{27} - \zeta_{12} q^{28} - q^{31} + q^{36} + \zeta_{12}^{3} q^{37} + \zeta_{12}^{4} q^{39} + \zeta_{12}^{3} q^{43} - \zeta_{12}^{3} q^{48} + \zeta_{12} q^{52} - 2 \zeta_{12} q^{57} - \zeta_{12}^{2} q^{61} + \zeta_{12}^{3} q^{63} - q^{64} + \zeta_{12}^{5} q^{67} + \zeta_{12}^{3} q^{73} + 2 \zeta_{12}^{4} q^{76} - \zeta_{12}^{2} q^{79} - \zeta_{12}^{2} q^{81} + q^{84} + \zeta_{12}^{4} q^{91} - \zeta_{12}^{5} q^{93} - \zeta_{12}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} + 2 q^{21} - 4 q^{31} + 4 q^{36} - 2 q^{39} - 4 q^{61} - 4 q^{64} - 4 q^{76} - 2 q^{79} - 2 q^{81} + 4 q^{84} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
824.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 −0.866025 + 0.500000i 0 0.500000 0.866025i 0
824.2 0 0.866025 0.500000i 0.500000 + 0.866025i 0 0 0.866025 0.500000i 0 0.500000 0.866025i 0
2024.1 0 −0.866025 0.500000i 0.500000 0.866025i 0 0 −0.866025 0.500000i 0 0.500000 + 0.866025i 0
2024.2 0 0.866025 + 0.500000i 0.500000 0.866025i 0 0 0.866025 + 0.500000i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner
185.n even 6 1 inner
555.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.x.a 4
3.b odd 2 1 CM 2775.1.x.a 4
5.b even 2 1 inner 2775.1.x.a 4
5.c odd 4 1 111.1.i.a 2
5.c odd 4 1 2775.1.z.a 2
15.d odd 2 1 inner 2775.1.x.a 4
15.e even 4 1 111.1.i.a 2
15.e even 4 1 2775.1.z.a 2
20.e even 4 1 1776.1.bw.a 2
37.c even 3 1 inner 2775.1.x.a 4
45.k odd 12 1 2997.1.l.a 2
45.k odd 12 1 2997.1.u.a 2
45.l even 12 1 2997.1.l.a 2
45.l even 12 1 2997.1.u.a 2
60.l odd 4 1 1776.1.bw.a 2
111.i odd 6 1 inner 2775.1.x.a 4
185.n even 6 1 inner 2775.1.x.a 4
185.s odd 12 1 111.1.i.a 2
185.s odd 12 1 2775.1.z.a 2
555.w odd 6 1 inner 2775.1.x.a 4
555.bi even 12 1 111.1.i.a 2
555.bi even 12 1 2775.1.z.a 2
740.bg even 12 1 1776.1.bw.a 2
1665.co odd 12 1 2997.1.u.a 2
1665.cp even 12 1 2997.1.u.a 2
1665.de odd 12 1 2997.1.l.a 2
1665.dg even 12 1 2997.1.l.a 2
2220.cu odd 12 1 1776.1.bw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.i.a 2 5.c odd 4 1
111.1.i.a 2 15.e even 4 1
111.1.i.a 2 185.s odd 12 1
111.1.i.a 2 555.bi even 12 1
1776.1.bw.a 2 20.e even 4 1
1776.1.bw.a 2 60.l odd 4 1
1776.1.bw.a 2 740.bg even 12 1
1776.1.bw.a 2 2220.cu odd 12 1
2775.1.x.a 4 1.a even 1 1 trivial
2775.1.x.a 4 3.b odd 2 1 CM
2775.1.x.a 4 5.b even 2 1 inner
2775.1.x.a 4 15.d odd 2 1 inner
2775.1.x.a 4 37.c even 3 1 inner
2775.1.x.a 4 111.i odd 6 1 inner
2775.1.x.a 4 185.n even 6 1 inner
2775.1.x.a 4 555.w odd 6 1 inner
2775.1.z.a 2 5.c odd 4 1
2775.1.z.a 2 15.e even 4 1
2775.1.z.a 2 185.s odd 12 1
2775.1.z.a 2 555.bi even 12 1
2997.1.l.a 2 45.k odd 12 1
2997.1.l.a 2 45.l even 12 1
2997.1.l.a 2 1665.de odd 12 1
2997.1.l.a 2 1665.dg even 12 1
2997.1.u.a 2 45.k odd 12 1
2997.1.u.a 2 45.l even 12 1
2997.1.u.a 2 1665.co odd 12 1
2997.1.u.a 2 1665.cp even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2775, [\chi])\).

Hecke characteristic polynomials

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