Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(417,1456)]
 
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, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 2) q^{7} + 3 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} - q^{13} + (7 \zeta_{6} - 7) q^{17} - 7 \zeta_{6} q^{19} - 6 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} - 5 q^{29} - 8 \zeta_{6} q^{37} - 2 q^{43} + 7 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + ( - 3 \zeta_{6} + 3) q^{53} + (7 \zeta_{6} - 7) q^{59} + 7 \zeta_{6} q^{61} + (3 \zeta_{6} - 9) q^{63} + (3 \zeta_{6} - 3) q^{67} + 5 q^{71} + (14 \zeta_{6} - 14) q^{73} + ( - 6 \zeta_{6} - 3) q^{77} - 6 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 3 \zeta_{6} + 2) q^{91} - 14 q^{97} - 9 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{7} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 7 q^{17} - 7 q^{19} - 6 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{37} - 4 q^{43} + 7 q^{47} - 13 q^{49} + 3 q^{53} - 7 q^{59} + 7 q^{61} - 15 q^{63} - 3 q^{67} + 10 q^{71} - 14 q^{73} - 12 q^{77} - 6 q^{79} - 9 q^{81} + q^{91} - 28 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −0.500000 + 2.59808i 0 1.50000 + 2.59808i 0
625.1 0 0 0 0 0 −0.500000 2.59808i 0 1.50000 2.59808i 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
7.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.r.g 2
4.b odd 2 1 91.2.e.a 2
7.c even 3 1 inner 1456.2.r.g 2
12.b even 2 1 819.2.j.b 2
28.d even 2 1 637.2.e.a 2
28.f even 6 1 637.2.a.d 1
28.f even 6 1 637.2.e.a 2
28.g odd 6 1 91.2.e.a 2
28.g odd 6 1 637.2.a.c 1
52.b odd 2 1 1183.2.e.b 2
84.j odd 6 1 5733.2.a.d 1
84.n even 6 1 819.2.j.b 2
84.n even 6 1 5733.2.a.c 1
364.x even 6 1 8281.2.a.e 1
364.bl odd 6 1 1183.2.e.b 2
364.bl odd 6 1 8281.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 4.b odd 2 1
91.2.e.a 2 28.g odd 6 1
637.2.a.c 1 28.g odd 6 1
637.2.a.d 1 28.f even 6 1
637.2.e.a 2 28.d even 2 1
637.2.e.a 2 28.f even 6 1
819.2.j.b 2 12.b even 2 1
819.2.j.b 2 84.n even 6 1
1183.2.e.b 2 52.b odd 2 1
1183.2.e.b 2 364.bl odd 6 1
1456.2.r.g 2 1.a even 1 1 trivial
1456.2.r.g 2 7.c even 3 1 inner
5733.2.a.c 1 84.n even 6 1
5733.2.a.d 1 84.j odd 6 1
8281.2.a.e 1 364.x even 6 1
8281.2.a.f 1 364.bl odd 6 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$71$ \( (T - 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T + 14)^{2} \) Copy content Toggle raw display
show more
show less