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Properties

Label 1344.2.bk.k
Level $1344$
Weight $2$
Character orbit 1344.bk
Analytic conductor $10.732$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(289,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.bk (of order \(6\), degree \(2\), 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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{3} + \beta_{2} + 1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{3} + \beta_{2} + 1) q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{7} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{5} + 4 q^{9} - 8 q^{17} + 6 q^{21} + 2 q^{25} + 2 q^{33} - 18 q^{37} + 8 q^{41} + 6 q^{45} + 20 q^{49} - 42 q^{53} + 36 q^{57} - 36 q^{61} + 28 q^{65} - 30 q^{73} - 54 q^{77} - 4 q^{81} - 16 q^{89} + 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{6} - 56\nu^{4} - 504\nu^{2} - 2025 ) / 1400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{6} - 224\nu^{4} - 616\nu^{2} - 2475 ) / 1400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} + 224\nu^{5} + 616\nu^{3} + 2475\nu ) / 7000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 1 ) / 56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 9\nu^{5} - 56\nu^{3} - 225\nu ) / 125 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - 4\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} - 4\beta_{4} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{3} + 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 56\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -56\beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -280\beta_{6} - 280\beta_{4} - \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.656712 + 2.13746i
1.52274 − 1.63746i
0.656712 − 2.13746i
−1.52274 + 1.63746i
−0.656712 − 2.13746i
1.52274 + 1.63746i
0.656712 + 2.13746i
−1.52274 − 1.63746i
0 −0.866025 + 0.500000i 0 −1.13746 − 0.656712i 0 2.17945 − 1.50000i 0 0.500000 − 0.866025i 0
289.2 0 −0.866025 + 0.500000i 0 2.63746 + 1.52274i 0 −2.17945 − 1.50000i 0 0.500000 − 0.866025i 0
289.3 0 0.866025 − 0.500000i 0 −1.13746 − 0.656712i 0 −2.17945 + 1.50000i 0 0.500000 − 0.866025i 0
289.4 0 0.866025 − 0.500000i 0 2.63746 + 1.52274i 0 2.17945 + 1.50000i 0 0.500000 − 0.866025i 0
865.1 0 −0.866025 − 0.500000i 0 −1.13746 + 0.656712i 0 2.17945 + 1.50000i 0 0.500000 + 0.866025i 0
865.2 0 −0.866025 − 0.500000i 0 2.63746 − 1.52274i 0 −2.17945 + 1.50000i 0 0.500000 + 0.866025i 0
865.3 0 0.866025 + 0.500000i 0 −1.13746 + 0.656712i 0 −2.17945 − 1.50000i 0 0.500000 + 0.866025i 0
865.4 0 0.866025 + 0.500000i 0 2.63746 − 1.52274i 0 2.17945 − 1.50000i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.k odd 6 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.bk.k yes 8
4.b odd 2 1 inner 1344.2.bk.k yes 8
7.c even 3 1 1344.2.bk.j ✓ 8
8.b even 2 1 1344.2.bk.j ✓ 8
8.d odd 2 1 1344.2.bk.j ✓ 8
28.g odd 6 1 1344.2.bk.j ✓ 8
56.k odd 6 1 inner 1344.2.bk.k yes 8
56.p even 6 1 inner 1344.2.bk.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.bk.j ✓ 8 7.c even 3 1
1344.2.bk.j ✓ 8 8.b even 2 1
1344.2.bk.j ✓ 8 8.d odd 2 1
1344.2.bk.j ✓ 8 28.g odd 6 1
1344.2.bk.k yes 8 1.a even 1 1 trivial
1344.2.bk.k yes 8 4.b odd 2 1 inner
1344.2.bk.k yes 8 56.k odd 6 1 inner
1344.2.bk.k yes 8 56.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{4} - 3T_{5}^{3} - T_{5}^{2} + 12T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 29T_{11}^{6} + 645T_{11}^{4} - 5684T_{11}^{2} + 38416 \) Copy content Toggle raw display
\( T_{23}^{4} + 12T_{23}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 3 T^{3} - T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 29 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$13$ \( (T^{4} + 23 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} - 69 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 11 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 19 T^{2} + 361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 9 T^{3} + 29 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 56)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 29 T^{2} + 196)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 92 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( (T^{4} + 21 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 269 T^{6} + \cdots + 221533456 \) Copy content Toggle raw display
$61$ \( (T^{4} + 18 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 257 T^{6} + \cdots + 268435456 \) Copy content Toggle raw display
$71$ \( (T^{4} - 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 15 T^{3} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 254 T^{6} + \cdots + 62742241 \) Copy content Toggle raw display
$83$ \( (T^{4} + 249 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 13 T - 86)^{4} \) Copy content Toggle raw display
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