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Properties

Label 2541.1.k.a
Level $2541$
Weight $1$
Character orbit 2541.k
Analytic conductor $1.268$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
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] = [2541,1,Mod(362,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.362");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2541.k (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: \(1.26812419710\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective 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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{2} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{7} + \zeta_{24}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{2} q^{3} - \zeta_{24}^{8} q^{4} - \zeta_{24}^{5} q^{7} + \zeta_{24}^{4} q^{9} - \zeta_{24}^{10} q^{12} + ( - \zeta_{24}^{7} + \zeta_{24}^{5}) q^{13} - \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{19} - \zeta_{24}^{7} q^{21} - \zeta_{24}^{8} q^{25} + \zeta_{24}^{6} q^{27} - \zeta_{24} q^{28} - \zeta_{24}^{2} q^{31} + q^{36} + (\zeta_{24}^{6} + \zeta_{24}^{2}) q^{37} + ( - \zeta_{24}^{9} + \zeta_{24}^{7}) q^{39} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{43} - \zeta_{24}^{6} q^{48} + \zeta_{24}^{10} q^{49} + ( - \zeta_{24}^{3} + \zeta_{24}) q^{52} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{57} + (\zeta_{24}^{5} + \zeta_{24}^{3}) q^{61} - \zeta_{24}^{9} q^{63} - q^{64} + ( - \zeta_{24}^{9} - \zeta_{24}^{7}) q^{73} - \zeta_{24}^{10} q^{75} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{76} + (\zeta_{24}^{7} - \zeta_{24}) q^{79} + \zeta_{24}^{8} q^{81} - \zeta_{24}^{3} q^{84} + ( - \zeta_{24}^{10} - 1) q^{91} - \zeta_{24}^{4} q^{93} + ( - \zeta_{24}^{8} - \zeta_{24}^{4}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 4 q^{9} - 4 q^{16} + 4 q^{25} + 8 q^{36} - 8 q^{64} - 4 q^{81} - 8 q^{91} - 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2541\mathbb{Z}\right)^\times\).

\(n\) \(848\) \(1816\) \(2059\)
\(\chi(n)\) \(-1\) \(\zeta_{24}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
362.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 −0.965926 − 0.258819i 0 0.500000 − 0.866025i 0
362.2 0 −0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 0.965926 + 0.258819i 0 0.500000 − 0.866025i 0
362.3 0 0.866025 − 0.500000i 0.500000 + 0.866025i 0 0 −0.258819 + 0.965926i 0 0.500000 − 0.866025i 0
362.4 0 0.866025 − 0.500000i 0.500000 + 0.866025i 0 0 0.258819 − 0.965926i 0 0.500000 − 0.866025i 0
1088.1 0 −0.866025 − 0.500000i 0.500000 − 0.866025i 0 0 −0.965926 + 0.258819i 0 0.500000 + 0.866025i 0
1088.2 0 −0.866025 − 0.500000i 0.500000 − 0.866025i 0 0 0.965926 − 0.258819i 0 0.500000 + 0.866025i 0
1088.3 0 0.866025 + 0.500000i 0.500000 − 0.866025i 0 0 −0.258819 − 0.965926i 0 0.500000 + 0.866025i 0
1088.4 0 0.866025 + 0.500000i 0.500000 − 0.866025i 0 0 0.258819 + 0.965926i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 362.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.d odd 6 1 inner
11.b odd 2 1 inner
21.g even 6 1 inner
33.d even 2 1 inner
77.i even 6 1 inner
231.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2541.1.k.a ✓ 8
3.b odd 2 1 CM 2541.1.k.a ✓ 8
7.d odd 6 1 inner 2541.1.k.a ✓ 8
11.b odd 2 1 inner 2541.1.k.a ✓ 8
11.c even 5 4 2541.1.bn.a 32
11.d odd 10 4 2541.1.bn.a 32
21.g even 6 1 inner 2541.1.k.a ✓ 8
33.d even 2 1 inner 2541.1.k.a ✓ 8
33.f even 10 4 2541.1.bn.a 32
33.h odd 10 4 2541.1.bn.a 32
77.i even 6 1 inner 2541.1.k.a ✓ 8
77.n even 30 4 2541.1.bn.a 32
77.p odd 30 4 2541.1.bn.a 32
231.k odd 6 1 inner 2541.1.k.a ✓ 8
231.bc even 30 4 2541.1.bn.a 32
231.bf odd 30 4 2541.1.bn.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2541.1.k.a ✓ 8 1.a even 1 1 trivial
2541.1.k.a ✓ 8 3.b odd 2 1 CM
2541.1.k.a ✓ 8 7.d odd 6 1 inner
2541.1.k.a ✓ 8 11.b odd 2 1 inner
2541.1.k.a ✓ 8 21.g even 6 1 inner
2541.1.k.a ✓ 8 33.d even 2 1 inner
2541.1.k.a ✓ 8 77.i even 6 1 inner
2541.1.k.a ✓ 8 231.k odd 6 1 inner
2541.1.bn.a 32 11.c even 5 4
2541.1.bn.a 32 11.d odd 10 4
2541.1.bn.a 32 33.f even 10 4
2541.1.bn.a 32 33.h odd 10 4
2541.1.bn.a 32 77.n even 30 4
2541.1.bn.a 32 77.p odd 30 4
2541.1.bn.a 32 231.bc even 30 4
2541.1.bn.a 32 231.bf odd 30 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2541, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
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