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Properties

Label 2548.1.ca.a
Level $2548$
Weight $1$
Character orbit 2548.ca
Analytic conductor $1.272$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,1,Mod(227,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 2, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.227");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2548.ca (of order \(12\), degree \(4\), 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: \(1.27161765219\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective 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.

\(f(q)\) \(=\) \( q + \zeta_{48}^{18} q^{2} - \zeta_{48}^{12} q^{4} + (\zeta_{48}^{19} + \zeta_{48}^{9}) q^{5} + \zeta_{48}^{6} q^{8} - \zeta_{48}^{16} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{48}^{18} q^{2} - \zeta_{48}^{12} q^{4} + (\zeta_{48}^{19} + \zeta_{48}^{9}) q^{5} + \zeta_{48}^{6} q^{8} - \zeta_{48}^{16} q^{9} + ( - \zeta_{48}^{13} - \zeta_{48}^{3}) q^{10} - \zeta_{48} q^{13} - q^{16} + (\zeta_{48}^{23} - \zeta_{48}) q^{17} + \zeta_{48}^{10} q^{18} + ( - \zeta_{48}^{21} + \zeta_{48}^{7}) q^{20} + (\zeta_{48}^{18} + \cdots - \zeta_{48}^{4}) q^{25} + \cdots + (\zeta_{48}^{23} + \zeta_{48}^{5}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 16 q^{16} + 8 q^{50} - 8 q^{58} + 16 q^{74} - 8 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(\zeta_{48}^{20}\) \(\zeta_{48}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
0.991445 + 0.130526i
−0.991445 − 0.130526i
−0.130526 + 0.991445i
0.130526 − 0.991445i
0.608761 + 0.793353i
−0.608761 − 0.793353i
−0.793353 + 0.608761i
0.793353 − 0.608761i
0.991445 − 0.130526i
−0.991445 + 0.130526i
−0.130526 − 0.991445i
0.130526 + 0.991445i
0.608761 − 0.793353i
−0.608761 + 0.793353i
−0.793353 − 0.608761i
0.793353 + 0.608761i
−0.707107 + 0.707107i 0 − 1.00000i −0.410670 + 1.53264i 0 0 0.707107 + 0.707107i 0.500000 − 0.866025i −0.793353 − 1.37413i
227.2 −0.707107 + 0.707107i 0 − 1.00000i 0.410670 − 1.53264i 0 0 0.707107 + 0.707107i 0.500000 − 0.866025i 0.793353 + 1.37413i
227.3 0.707107 − 0.707107i 0 − 1.00000i −0.315118 + 1.17604i 0 0 −0.707107 − 0.707107i 0.500000 − 0.866025i 0.608761 + 1.05441i
227.4 0.707107 − 0.707107i 0 − 1.00000i 0.315118 − 1.17604i 0 0 −0.707107 − 0.707107i 0.500000 − 0.866025i −0.608761 − 1.05441i
423.1 −0.707107 − 0.707107i 0 1.00000i −0.252157 − 0.0675653i 0 0 0.707107 − 0.707107i 0.500000 − 0.866025i 0.130526 + 0.226078i
423.2 −0.707107 − 0.707107i 0 1.00000i 0.252157 + 0.0675653i 0 0 0.707107 − 0.707107i 0.500000 − 0.866025i −0.130526 − 0.226078i
423.3 0.707107 + 0.707107i 0 1.00000i −1.91532 − 0.513210i 0 0 −0.707107 + 0.707107i 0.500000 − 0.866025i −0.991445 − 1.71723i
423.4 0.707107 + 0.707107i 0 1.00000i 1.91532 + 0.513210i 0 0 −0.707107 + 0.707107i 0.500000 − 0.866025i 0.991445 + 1.71723i
999.1 −0.707107 − 0.707107i 0 1.00000i −0.410670 − 1.53264i 0 0 0.707107 − 0.707107i 0.500000 + 0.866025i −0.793353 + 1.37413i
999.2 −0.707107 − 0.707107i 0 1.00000i 0.410670 + 1.53264i 0 0 0.707107 − 0.707107i 0.500000 + 0.866025i 0.793353 − 1.37413i
999.3 0.707107 + 0.707107i 0 1.00000i −0.315118 − 1.17604i 0 0 −0.707107 + 0.707107i 0.500000 + 0.866025i 0.608761 − 1.05441i
999.4 0.707107 + 0.707107i 0 1.00000i 0.315118 + 1.17604i 0 0 −0.707107 + 0.707107i 0.500000 + 0.866025i −0.608761 + 1.05441i
1783.1 −0.707107 + 0.707107i 0 − 1.00000i −0.252157 + 0.0675653i 0 0 0.707107 + 0.707107i 0.500000 + 0.866025i 0.130526 − 0.226078i
1783.2 −0.707107 + 0.707107i 0 − 1.00000i 0.252157 − 0.0675653i 0 0 0.707107 + 0.707107i 0.500000 + 0.866025i −0.130526 + 0.226078i
1783.3 0.707107 − 0.707107i 0 − 1.00000i −1.91532 + 0.513210i 0 0 −0.707107 − 0.707107i 0.500000 + 0.866025i −0.991445 + 1.71723i
1783.4 0.707107 − 0.707107i 0 − 1.00000i 1.91532 − 0.513210i 0 0 −0.707107 − 0.707107i 0.500000 + 0.866025i 0.991445 − 1.71723i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.b odd 2 1 inner
28.d even 2 1 inner
91.x odd 12 1 inner
91.ba even 12 1 inner
364.bz odd 12 1 inner
364.ca even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.1.ca.a 16
4.b odd 2 1 CM 2548.1.ca.a 16
7.b odd 2 1 inner 2548.1.ca.a 16
7.c even 3 1 2548.1.bw.a ✓ 16
7.c even 3 1 2548.1.ch.a 16
7.d odd 6 1 2548.1.bw.a ✓ 16
7.d odd 6 1 2548.1.ch.a 16
13.f odd 12 1 2548.1.ch.a 16
28.d even 2 1 inner 2548.1.ca.a 16
28.f even 6 1 2548.1.bw.a ✓ 16
28.f even 6 1 2548.1.ch.a 16
28.g odd 6 1 2548.1.bw.a ✓ 16
28.g odd 6 1 2548.1.ch.a 16
52.l even 12 1 2548.1.ch.a 16
91.w even 12 1 2548.1.bw.a ✓ 16
91.x odd 12 1 inner 2548.1.ca.a 16
91.ba even 12 1 inner 2548.1.ca.a 16
91.bc even 12 1 2548.1.ch.a 16
91.bd odd 12 1 2548.1.bw.a ✓ 16
364.bt even 12 1 2548.1.bw.a ✓ 16
364.bv odd 12 1 2548.1.ch.a 16
364.bz odd 12 1 inner 2548.1.ca.a 16
364.ca even 12 1 inner 2548.1.ca.a 16
364.cg odd 12 1 2548.1.bw.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2548.1.bw.a ✓ 16 7.c even 3 1
2548.1.bw.a ✓ 16 7.d odd 6 1
2548.1.bw.a ✓ 16 28.f even 6 1
2548.1.bw.a ✓ 16 28.g odd 6 1
2548.1.bw.a ✓ 16 91.w even 12 1
2548.1.bw.a ✓ 16 91.bd odd 12 1
2548.1.bw.a ✓ 16 364.bt even 12 1
2548.1.bw.a ✓ 16 364.cg odd 12 1
2548.1.ca.a 16 1.a even 1 1 trivial
2548.1.ca.a 16 4.b odd 2 1 CM
2548.1.ca.a 16 7.b odd 2 1 inner
2548.1.ca.a 16 28.d even 2 1 inner
2548.1.ca.a 16 91.x odd 12 1 inner
2548.1.ca.a 16 91.ba even 12 1 inner
2548.1.ca.a 16 364.bz odd 12 1 inner
2548.1.ca.a 16 364.ca even 12 1 inner
2548.1.ch.a 16 7.c even 3 1
2548.1.ch.a 16 7.d odd 6 1
2548.1.ch.a 16 13.f odd 12 1
2548.1.ch.a 16 28.f even 6 1
2548.1.ch.a 16 28.g odd 6 1
2548.1.ch.a 16 52.l even 12 1
2548.1.ch.a 16 91.bc even 12 1
2548.1.ch.a 16 364.bv odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2548, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$17$ \( (T^{8} - 8 T^{6} + 20 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} - 12 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + 3 T^{2} + 9)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 12 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 12 T^{4} + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 12 T^{12} + \cdots + 16 \) Copy content Toggle raw display
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