Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(223,1344)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.p (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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1) q^{7} - q^{9} - 2 \beta_{3} q^{11} + 4 q^{13} + 2 \beta_{2} q^{17} - 4 \beta_1 q^{19} + ( - \beta_{2} - 1) q^{21} - 6 \beta_1 q^{23} - 5 q^{25} - \beta_1 q^{27} - 2 \beta_{2} q^{29} + 2 \beta_{3} q^{31} - 2 \beta_{2} q^{33} - 4 \beta_{2} q^{37} + 4 \beta_1 q^{39} + 2 \beta_{2} q^{41} - 4 \beta_{3} q^{43} - 4 \beta_{3} q^{47} + ( - 2 \beta_{2} + 5) q^{49} - 2 \beta_{3} q^{51} - 2 \beta_{2} q^{53} + 4 q^{57} - 12 \beta_1 q^{59} - 4 q^{61} + (\beta_{3} - \beta_1) q^{63} + 6 q^{69} + 6 \beta_1 q^{71} - 5 \beta_1 q^{75} + ( - 2 \beta_{2} + 12) q^{77} + 2 \beta_1 q^{79} + q^{81} + 12 \beta_1 q^{83} + 2 \beta_{3} q^{87} + 2 \beta_{2} q^{89} + ( - 4 \beta_{3} + 4 \beta_1) q^{91} + 2 \beta_{2} q^{93} - 4 \beta_{2} q^{97} + 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 16 q^{13} - 4 q^{21} - 20 q^{25} + 20 q^{49} + 16 q^{57} - 16 q^{61} + 24 q^{69} + 48 q^{77} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
0 1.00000i 0 0 0 −2.44949 1.00000i 0 −1.00000 0
223.2 0 1.00000i 0 0 0 2.44949 1.00000i 0 −1.00000 0
223.3 0 1.00000i 0 0 0 −2.44949 + 1.00000i 0 −1.00000 0
223.4 0 1.00000i 0 0 0 2.44949 + 1.00000i 0 −1.00000 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
4.b odd 2 1 inner
56.e even 2 1 inner
56.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.p.b yes 4
3.b odd 2 1 4032.2.p.d 4
4.b odd 2 1 inner 1344.2.p.b yes 4
7.b odd 2 1 1344.2.p.a 4
8.b even 2 1 1344.2.p.a 4
8.d odd 2 1 1344.2.p.a 4
12.b even 2 1 4032.2.p.d 4
21.c even 2 1 4032.2.p.a 4
24.f even 2 1 4032.2.p.a 4
24.h odd 2 1 4032.2.p.a 4
28.d even 2 1 1344.2.p.a 4
56.e even 2 1 inner 1344.2.p.b yes 4
56.h odd 2 1 inner 1344.2.p.b yes 4
84.h odd 2 1 4032.2.p.a 4
168.e odd 2 1 4032.2.p.d 4
168.i even 2 1 4032.2.p.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.a 4 7.b odd 2 1
1344.2.p.a 4 8.b even 2 1
1344.2.p.a 4 8.d odd 2 1
1344.2.p.a 4 28.d even 2 1
1344.2.p.b yes 4 1.a even 1 1 trivial
1344.2.p.b yes 4 4.b odd 2 1 inner
1344.2.p.b yes 4 56.e even 2 1 inner
1344.2.p.b yes 4 56.h odd 2 1 inner
4032.2.p.a 4 21.c even 2 1
4032.2.p.a 4 24.f even 2 1
4032.2.p.a 4 24.h odd 2 1
4032.2.p.a 4 84.h odd 2 1
4032.2.p.d 4 3.b odd 2 1
4032.2.p.d 4 12.b even 2 1
4032.2.p.d 4 168.e odd 2 1
4032.2.p.d 4 168.i even 2 1

Hecke kernels

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

\( T_{5} \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 10T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
show more
show less