Properties

Label 2736.1.b.b
Level $2736$
Weight $1$
Character orbit 2736.b
Analytic conductor $1.365$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -19
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,1,Mod(2735,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2735");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2736.b (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: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{24}^{7} - \zeta_{24}^{5}) q^{5} - \zeta_{24}^{6} q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{24}^{7} - \zeta_{24}^{5}) q^{5} - \zeta_{24}^{6} q^{7} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{11} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{17} + \zeta_{24}^{6} q^{19} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{23} + (\zeta_{24}^{10} - \zeta_{24}^{2} - 1) q^{25} + (\zeta_{24}^{11} - \zeta_{24}) q^{35} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{43} + (\zeta_{24}^{7} - \zeta_{24}^{5}) q^{47} + ( - \zeta_{24}^{8} + \cdots - \zeta_{24}^{4}) q^{55} + \cdots + ( - \zeta_{24}^{11} + \zeta_{24}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} - 8 q^{73} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2735.1
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 0.258819i
0 0 0 1.93185i 0 1.00000i 0 0 0
2735.2 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.3 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.4 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.5 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.6 0 0 0 0.517638i 0 1.00000i 0 0 0
2735.7 0 0 0 1.93185i 0 1.00000i 0 0 0
2735.8 0 0 0 1.93185i 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2735.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
57.d even 2 1 inner
76.d even 2 1 inner
228.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.b.b 8
3.b odd 2 1 inner 2736.1.b.b 8
4.b odd 2 1 inner 2736.1.b.b 8
12.b even 2 1 inner 2736.1.b.b 8
19.b odd 2 1 CM 2736.1.b.b 8
57.d even 2 1 inner 2736.1.b.b 8
76.d even 2 1 inner 2736.1.b.b 8
228.b odd 2 1 inner 2736.1.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.1.b.b 8 1.a even 1 1 trivial
2736.1.b.b 8 3.b odd 2 1 inner
2736.1.b.b 8 4.b odd 2 1 inner
2736.1.b.b 8 12.b even 2 1 inner
2736.1.b.b 8 19.b odd 2 1 CM
2736.1.b.b 8 57.d even 2 1 inner
2736.1.b.b 8 76.d even 2 1 inner
2736.1.b.b 8 228.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 4T_{5}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2736, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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