Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(1247,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.1247");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.j (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{3} - \beta_{7} q^{5} + \beta_1 q^{7} + ( - \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{3} - \beta_{7} q^{5} + \beta_1 q^{7} + ( - \beta_{6} + 2) q^{9} - 4 \beta_{4} q^{11} + ( - \beta_{7} - 2 \beta_{2}) q^{13} + ( - \beta_{3} + \beta_1) q^{15} + 2 \beta_{6} q^{17} + ( - 2 \beta_{5} + \beta_{4}) q^{19} - \beta_{2} q^{21} - 4 \beta_{3} q^{23} - 3 q^{25} + ( - \beta_{5} + 4 \beta_{4}) q^{27} - 2 \beta_{7} q^{29} + (4 \beta_{6} + 4) q^{33} + \beta_{4} q^{35} + ( - 2 \beta_{7} - 4 \beta_{2}) q^{37} + (\beta_{3} + 5 \beta_1) q^{39} - 2 \beta_{6} q^{41} + ( - 4 \beta_{5} + 2 \beta_{4}) q^{43} + ( - 3 \beta_{7} - 2 \beta_{2}) q^{45} + 4 \beta_{3} q^{47} - q^{49} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{51} + 4 \beta_{7} q^{53} - 8 \beta_1 q^{55} + ( - \beta_{6} + 5) q^{57} - 3 \beta_{4} q^{59} + ( - 3 \beta_{7} - 6 \beta_{2}) q^{61} + (\beta_{3} + 2 \beta_1) q^{63} - 2 \beta_{6} q^{65} + ( - 8 \beta_{5} + 4 \beta_{4}) q^{67} + ( - 12 \beta_{7} - 4 \beta_{2}) q^{69} + 6 \beta_{3} q^{71} - 14 q^{73} + (3 \beta_{5} - 3 \beta_{4}) q^{75} - 4 \beta_{7} q^{77} + ( - 4 \beta_{6} - 1) q^{81} - 5 \beta_{4} q^{83} + (2 \beta_{7} + 4 \beta_{2}) q^{85} + ( - 2 \beta_{3} + 2 \beta_1) q^{87} + 2 \beta_{6} q^{89} + (2 \beta_{5} - \beta_{4}) q^{91} - 2 \beta_{3} q^{95} + 2 q^{97} + ( - 8 \beta_{5} - 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} - 24 q^{25} + 32 q^{33} - 8 q^{49} + 40 q^{57} - 112 q^{73} - 8 q^{81} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{7} - 5\beta_{5} + 8\beta_{4} - 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{7} + 13\beta_{5} + 8\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1247.1
1.14412 + 1.14412i
0.437016 0.437016i
1.14412 1.14412i
0.437016 + 0.437016i
−0.437016 0.437016i
−1.14412 + 1.14412i
−0.437016 + 0.437016i
−1.14412 1.14412i
0 −1.58114 0.707107i 0 −1.41421 0 1.00000i 0 2.00000 + 2.23607i 0
1247.2 0 −1.58114 0.707107i 0 1.41421 0 1.00000i 0 2.00000 + 2.23607i 0
1247.3 0 −1.58114 + 0.707107i 0 −1.41421 0 1.00000i 0 2.00000 2.23607i 0
1247.4 0 −1.58114 + 0.707107i 0 1.41421 0 1.00000i 0 2.00000 2.23607i 0
1247.5 0 1.58114 0.707107i 0 −1.41421 0 1.00000i 0 2.00000 2.23607i 0
1247.6 0 1.58114 0.707107i 0 1.41421 0 1.00000i 0 2.00000 2.23607i 0
1247.7 0 1.58114 + 0.707107i 0 −1.41421 0 1.00000i 0 2.00000 + 2.23607i 0
1247.8 0 1.58114 + 0.707107i 0 1.41421 0 1.00000i 0 2.00000 + 2.23607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1247.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.2.j.h 8
3.b odd 2 1 inner 1344.2.j.h 8
4.b odd 2 1 inner 1344.2.j.h 8
8.b even 2 1 inner 1344.2.j.h 8
8.d odd 2 1 inner 1344.2.j.h 8
12.b even 2 1 inner 1344.2.j.h 8
24.f even 2 1 inner 1344.2.j.h 8
24.h odd 2 1 inner 1344.2.j.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.j.h 8 1.a even 1 1 trivial
1344.2.j.h 8 3.b odd 2 1 inner
1344.2.j.h 8 4.b odd 2 1 inner
1344.2.j.h 8 8.b even 2 1 inner
1344.2.j.h 8 8.d odd 2 1 inner
1344.2.j.h 8 12.b even 2 1 inner
1344.2.j.h 8 24.f even 2 1 inner
1344.2.j.h 8 24.h odd 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 10 \) Copy content Toggle raw display
\( T_{23}^{2} - 80 \) Copy content Toggle raw display
\( T_{47}^{2} - 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 10)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 160)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 180)^{4} \) Copy content Toggle raw display
$73$ \( (T + 14)^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$97$ \( (T - 2)^{8} \) Copy content Toggle raw display
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