↑

Properties

Label 1456.2.p.a
Level $1456$
Weight $2$
Character orbit 1456.p
Analytic conductor $11.626$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1456,2,Mod(1455,1456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1456.1455"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,-6,0,-4,0,2,0,12,0,-2,0,12,0,0,0,0,0,8,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6262185343\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} - 3 q^{5} + (\beta - 2) q^{7} + q^{9} + 6 q^{11} + (2 \beta - 1) q^{13} + 6 q^{15} + 2 \beta q^{17} - \beta q^{19} + ( - 2 \beta + 4) q^{21} + 5 \beta q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} + \cdots + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 6 q^{5} - 4 q^{7} + 2 q^{9} + 12 q^{11} - 2 q^{13} + 12 q^{15} + 8 q^{21} + 8 q^{25} + 8 q^{27} - 18 q^{29} - 24 q^{33} + 12 q^{35} + 4 q^{39} - 12 q^{41} - 6 q^{45} + 2 q^{49} + 6 q^{53}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1455.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −2.00000 0 −3.00000 0 −2.00000 − 1.73205i 0 1.00000 0
1455.2 0 −2.00000 0 −3.00000 0 −2.00000 + 1.73205i 0 1.00000 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
364.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.p.a ✓ 2
4.b odd 2 1 1456.2.p.c yes 2
7.b odd 2 1 1456.2.p.d yes 2
13.b even 2 1 1456.2.p.b yes 2
28.d even 2 1 1456.2.p.b yes 2
52.b odd 2 1 1456.2.p.d yes 2
91.b odd 2 1 1456.2.p.c yes 2
364.h even 2 1 inner 1456.2.p.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1456.2.p.a ✓ 2 1.a even 1 1 trivial
1456.2.p.a ✓ 2 364.h even 2 1 inner
1456.2.p.b yes 2 13.b even 2 1
1456.2.p.b yes 2 28.d even 2 1
1456.2.p.c yes 2 4.b odd 2 1
1456.2.p.c yes 2 91.b odd 2 1
1456.2.p.d yes 2 7.b odd 2 1
1456.2.p.d yes 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 12 \) Copy content Toggle raw display
$19$ \( T^{2} + 3 \) Copy content Toggle raw display
$23$ \( T^{2} + 75 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 27 \) Copy content Toggle raw display
$37$ \( T^{2} + 12 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 27 \) Copy content Toggle raw display
$47$ \( T^{2} + 75 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 108 \) Copy content Toggle raw display
$61$ \( T^{2} + 48 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 3 \) Copy content Toggle raw display
$83$ \( T^{2} + 3 \) Copy content Toggle raw display
$89$ \( (T + 3)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
show more
show less