Properties

Label 1344.2.bb.a
Level $1344$
Weight $2$
Character orbit 1344.bb
Analytic conductor $10.732$
Analytic rank $0$
Dimension $4$
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(31,1344)]
 
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([3, 3, 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("1344.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.bb (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + (3 \zeta_{12}^{2} - 3) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + (3 \zeta_{12}^{2} - 3) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{11} + 2 q^{13} + 3 \zeta_{12}^{3} q^{15} - 8 \zeta_{12} q^{19} + ( - 3 \zeta_{12}^{2} + 1) q^{21} + 6 \zeta_{12} q^{23} - 4 \zeta_{12}^{2} q^{25} - \zeta_{12}^{3} q^{27} + (6 \zeta_{12}^{2} - 3) q^{29} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{31} + ( - 3 \zeta_{12}^{2} - 3) q^{33} + (3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{35} + ( - 4 \zeta_{12}^{2} - 4) q^{37} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{39} + ( - 12 \zeta_{12}^{2} + 6) q^{41} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{43} + 3 \zeta_{12}^{2} q^{45} + ( - 5 \zeta_{12}^{2} - 3) q^{49} + (3 \zeta_{12}^{2} - 6) q^{53} + ( - 9 \zeta_{12}^{3} + 18 \zeta_{12}) q^{55} - 8 q^{57} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{59} + ( - 8 \zeta_{12}^{2} + 8) q^{61} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{63} + (6 \zeta_{12}^{2} - 6) q^{65} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{67} + 6 q^{69} - 12 \zeta_{12}^{3} q^{71} + ( - 8 \zeta_{12}^{2} + 16) q^{73} - 4 \zeta_{12} q^{75} + (3 \zeta_{12}^{2} - 15) q^{77} + 5 \zeta_{12} q^{79} - \zeta_{12}^{2} q^{81} + 9 \zeta_{12}^{3} q^{83} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{87} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}) q^{91} + ( - 5 \zeta_{12}^{2} - 5) q^{93} + ( - 24 \zeta_{12}^{3} + 24 \zeta_{12}) q^{95} + ( - 6 \zeta_{12}^{2} + 3) q^{97} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} + 2 q^{9} + 8 q^{13} - 2 q^{21} - 8 q^{25} - 18 q^{33} - 24 q^{37} + 6 q^{45} - 22 q^{49} - 18 q^{53} - 32 q^{57} + 16 q^{61} - 12 q^{65} + 24 q^{69} + 48 q^{73} - 54 q^{77} - 2 q^{81} - 30 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 0.500000i 0 −1.50000 2.59808i 0 −0.866025 2.50000i 0 0.500000 + 0.866025i 0
31.2 0 0.866025 + 0.500000i 0 −1.50000 2.59808i 0 0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
607.1 0 −0.866025 + 0.500000i 0 −1.50000 + 2.59808i 0 −0.866025 + 2.50000i 0 0.500000 0.866025i 0
607.2 0 0.866025 0.500000i 0 −1.50000 + 2.59808i 0 0.866025 2.50000i 0 0.500000 0.866025i 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.j odd 6 1 inner
56.m 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.bb.a 4
4.b odd 2 1 inner 1344.2.bb.a 4
7.d odd 6 1 1344.2.bb.d yes 4
8.b even 2 1 1344.2.bb.d yes 4
8.d odd 2 1 1344.2.bb.d yes 4
28.f even 6 1 1344.2.bb.d yes 4
56.j odd 6 1 inner 1344.2.bb.a 4
56.m even 6 1 inner 1344.2.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.bb.a 4 1.a even 1 1 trivial
1344.2.bb.a 4 4.b odd 2 1 inner
1344.2.bb.a 4 56.j odd 6 1 inner
1344.2.bb.a 4 56.m even 6 1 inner
1344.2.bb.d yes 4 7.d odd 6 1
1344.2.bb.d yes 4 8.b even 2 1
1344.2.bb.d yes 4 8.d odd 2 1
1344.2.bb.d yes 4 28.f even 6 1

Hecke kernels

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

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$83$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
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