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Properties

Label 1456.2.s.g
Level $1456$
Weight $2$
Character orbit 1456.s
Analytic conductor $11.626$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1456,2,Mod(113,1456)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1456, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) 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("1456.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6262185343\)
Analytic rank: \(0\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{3} - 3 q^{5} + \zeta_{6} q^{7} - 6 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{11} + (3 \zeta_{6} - 4) q^{13} + (9 \zeta_{6} - 9) q^{15} - 2 \zeta_{6} q^{17} - 4 \zeta_{6} q^{19} + \cdots - 24 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 6 q^{5} + q^{7} - 6 q^{9} + 4 q^{11} - 5 q^{13} - 9 q^{15} - 2 q^{17} - 4 q^{19} + 6 q^{21} - q^{23} + 8 q^{25} - 18 q^{27} + 2 q^{29} - 20 q^{31} - 12 q^{33} - 3 q^{35} - 4 q^{37} + 6 q^{39}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 1.50000 − 2.59808i 0 −3.00000 0 0.500000 + 0.866025i 0 −3.00000 − 5.19615i 0
1121.1 0 1.50000 + 2.59808i 0 −3.00000 0 0.500000 − 0.866025i 0 −3.00000 + 5.19615i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1456.2.s.g 2
4.b odd 2 1 182.2.g.a ✓ 2
12.b even 2 1 1638.2.r.n 2
13.c even 3 1 inner 1456.2.s.g 2
28.d even 2 1 1274.2.g.k 2
28.f even 6 1 1274.2.e.l 2
28.f even 6 1 1274.2.h.c 2
28.g odd 6 1 1274.2.e.a 2
28.g odd 6 1 1274.2.h.n 2
52.i odd 6 1 2366.2.a.p 1
52.j odd 6 1 182.2.g.a ✓ 2
52.j odd 6 1 2366.2.a.g 1
52.l even 12 2 2366.2.d.h 2
156.p even 6 1 1638.2.r.n 2
364.q odd 6 1 1274.2.e.a 2
364.v even 6 1 1274.2.g.k 2
364.ba even 6 1 1274.2.h.c 2
364.bi odd 6 1 1274.2.h.n 2
364.br even 6 1 1274.2.e.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.a ✓ 2 4.b odd 2 1
182.2.g.a ✓ 2 52.j odd 6 1
1274.2.e.a 2 28.g odd 6 1
1274.2.e.a 2 364.q odd 6 1
1274.2.e.l 2 28.f even 6 1
1274.2.e.l 2 364.br even 6 1
1274.2.g.k 2 28.d even 2 1
1274.2.g.k 2 364.v even 6 1
1274.2.h.c 2 28.f even 6 1
1274.2.h.c 2 364.ba even 6 1
1274.2.h.n 2 28.g odd 6 1
1274.2.h.n 2 364.bi odd 6 1
1456.2.s.g 2 1.a even 1 1 trivial
1456.2.s.g 2 13.c even 3 1 inner
1638.2.r.n 2 12.b even 2 1
1638.2.r.n 2 156.p even 6 1
2366.2.a.g 1 52.j odd 6 1
2366.2.a.p 1 52.i odd 6 1
2366.2.d.h 2 52.l even 12 2

Hecke kernels

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

\( T_{3}^{2} - 3T_{3} + 9 \) 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
$3$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 10)^{2} \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$61$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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