Properties

Label 1456.2.s.e
Level $1456$
Weight $2$
Character orbit 1456.s
Analytic conductor $11.626$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(113,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + 3 q^{5} + \zeta_{6} q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + 3 q^{5} + \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 4) q^{13} + ( - 3 \zeta_{6} + 3) q^{15} - 6 \zeta_{6} q^{17} - 4 \zeta_{6} q^{19} + q^{21} + ( - 3 \zeta_{6} + 3) q^{23} + 4 q^{25} + 5 q^{27} + (6 \zeta_{6} - 6) q^{29} + 10 q^{31} + 3 \zeta_{6} q^{35} + (8 \zeta_{6} - 8) q^{37} + ( - 4 \zeta_{6} + 1) q^{39} + 8 \zeta_{6} q^{43} + 6 \zeta_{6} q^{45} - 6 q^{47} + (\zeta_{6} - 1) q^{49} - 6 q^{51} + 12 q^{53} - 4 q^{57} + 3 \zeta_{6} q^{59} - 11 \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{63} + ( - 9 \zeta_{6} + 12) q^{65} + ( - 2 \zeta_{6} + 2) q^{67} - 3 \zeta_{6} q^{69} - 3 \zeta_{6} q^{71} + 2 q^{73} + ( - 4 \zeta_{6} + 4) q^{75} + 4 q^{79} + (\zeta_{6} - 1) q^{81} - 18 \zeta_{6} q^{85} + 6 \zeta_{6} q^{87} + ( - 6 \zeta_{6} + 6) q^{89} + (\zeta_{6} + 3) q^{91} + ( - 10 \zeta_{6} + 10) q^{93} - 12 \zeta_{6} q^{95} - 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{5} + q^{7} + 2 q^{9} + 5 q^{13} + 3 q^{15} - 6 q^{17} - 4 q^{19} + 2 q^{21} + 3 q^{23} + 8 q^{25} + 10 q^{27} - 6 q^{29} + 20 q^{31} + 3 q^{35} - 8 q^{37} - 2 q^{39} + 8 q^{43} + 6 q^{45} - 12 q^{47} - q^{49} - 12 q^{51} + 24 q^{53} - 8 q^{57} + 3 q^{59} - 11 q^{61} - 2 q^{63} + 15 q^{65} + 2 q^{67} - 3 q^{69} - 3 q^{71} + 4 q^{73} + 4 q^{75} + 8 q^{79} - q^{81} - 18 q^{85} + 6 q^{87} + 6 q^{89} + 7 q^{91} + 10 q^{93} - 12 q^{95} - 2 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 3.00000 0 0.500000 + 0.866025i 0 1.00000 + 1.73205i 0
1121.1 0 0.500000 + 0.866025i 0 3.00000 0 0.500000 0.866025i 0 1.00000 1.73205i 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.e 2
4.b odd 2 1 182.2.g.b 2
12.b even 2 1 1638.2.r.c 2
13.c even 3 1 inner 1456.2.s.e 2
28.d even 2 1 1274.2.g.g 2
28.f even 6 1 1274.2.e.i 2
28.f even 6 1 1274.2.h.i 2
28.g odd 6 1 1274.2.e.d 2
28.g odd 6 1 1274.2.h.j 2
52.i odd 6 1 2366.2.a.n 1
52.j odd 6 1 182.2.g.b 2
52.j odd 6 1 2366.2.a.f 1
52.l even 12 2 2366.2.d.f 2
156.p even 6 1 1638.2.r.c 2
364.q odd 6 1 1274.2.e.d 2
364.v even 6 1 1274.2.g.g 2
364.ba even 6 1 1274.2.h.i 2
364.bi odd 6 1 1274.2.h.j 2
364.br even 6 1 1274.2.e.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 4.b odd 2 1
182.2.g.b 2 52.j odd 6 1
1274.2.e.d 2 28.g odd 6 1
1274.2.e.d 2 364.q odd 6 1
1274.2.e.i 2 28.f even 6 1
1274.2.e.i 2 364.br even 6 1
1274.2.g.g 2 28.d even 2 1
1274.2.g.g 2 364.v even 6 1
1274.2.h.i 2 28.f even 6 1
1274.2.h.i 2 364.ba even 6 1
1274.2.h.j 2 28.g odd 6 1
1274.2.h.j 2 364.bi odd 6 1
1456.2.s.e 2 1.a even 1 1 trivial
1456.2.s.e 2 13.c even 3 1 inner
1638.2.r.c 2 12.b even 2 1
1638.2.r.c 2 156.p even 6 1
2366.2.a.f 1 52.j odd 6 1
2366.2.a.n 1 52.i odd 6 1
2366.2.d.f 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 \(S_{2}^{\mathrm{new}}(1456, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) 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
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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