Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(193,1344)]
 
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, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (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: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} + 2 \beta_1 - 1) q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} - \beta_{2} q^{9} + (2 \beta_{3} + \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{13} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{15} + ( - 4 \beta_{2} + 4) q^{17} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{19} + ( - \beta_{2} + \beta_1 + 1) q^{21} - 4 \beta_{2} q^{23} + (2 \beta_{3} + 10 \beta_{2} - \beta_1 - 9) q^{25} + q^{27} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{29} + ( - \beta_{2} + 1) q^{31} + ( - \beta_{3} + 2 \beta_1 - 1) q^{33} + (\beta_{3} - 5 \beta_{2} - 3 \beta_1 + 11) q^{35} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{37} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{39} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{41}+ \cdots + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - q^{5} - 6 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - q^{5} - 6 q^{7} - 2 q^{9} - q^{11} - 10 q^{13} + 2 q^{15} + 8 q^{17} - 5 q^{19} + 3 q^{21} - 8 q^{23} - 19 q^{25} + 4 q^{27} - 6 q^{29} + 2 q^{31} - q^{33} + 30 q^{35} + 3 q^{37} + 5 q^{39} + 12 q^{41} + 14 q^{43} - q^{45} + 12 q^{47} - 10 q^{49} + 8 q^{51} + 11 q^{53} + 58 q^{55} + 10 q^{57} + 5 q^{59} + 20 q^{61} + 3 q^{63} - 26 q^{65} + 7 q^{67} + 16 q^{69} + 8 q^{71} + q^{73} - 19 q^{75} - 27 q^{77} - 8 q^{79} - 2 q^{81} - 14 q^{83} - 8 q^{85} + 3 q^{87} + 6 q^{89} + 15 q^{91} + 2 q^{93} + 26 q^{95} + 50 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu + 5 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} + 4\beta _1 + 5 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0 −0.500000 + 0.866025i 0 −2.13746 3.70219i 0 −1.50000 + 2.17945i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 1.63746 + 2.83616i 0 −1.50000 2.17945i 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 −2.13746 + 3.70219i 0 −1.50000 2.17945i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 1.63746 2.83616i 0 −1.50000 + 2.17945i 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
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 1344.2.q.w 4
4.b odd 2 1 1344.2.q.x 4
7.c even 3 1 inner 1344.2.q.w 4
7.c even 3 1 9408.2.a.ec 2
7.d odd 6 1 9408.2.a.dj 2
8.b even 2 1 168.2.q.c 4
8.d odd 2 1 336.2.q.g 4
24.f even 2 1 1008.2.s.r 4
24.h odd 2 1 504.2.s.i 4
28.f even 6 1 9408.2.a.dw 2
28.g odd 6 1 1344.2.q.x 4
28.g odd 6 1 9408.2.a.dp 2
56.e even 2 1 2352.2.q.bf 4
56.h odd 2 1 1176.2.q.l 4
56.j odd 6 1 1176.2.a.n 2
56.j odd 6 1 1176.2.q.l 4
56.k odd 6 1 336.2.q.g 4
56.k odd 6 1 2352.2.a.bf 2
56.m even 6 1 2352.2.a.ba 2
56.m even 6 1 2352.2.q.bf 4
56.p even 6 1 168.2.q.c 4
56.p even 6 1 1176.2.a.k 2
168.i even 2 1 3528.2.s.bk 4
168.s odd 6 1 504.2.s.i 4
168.s odd 6 1 3528.2.a.bk 2
168.v even 6 1 1008.2.s.r 4
168.v even 6 1 7056.2.a.cu 2
168.ba even 6 1 3528.2.a.bd 2
168.ba even 6 1 3528.2.s.bk 4
168.be odd 6 1 7056.2.a.ch 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 8.b even 2 1
168.2.q.c 4 56.p even 6 1
336.2.q.g 4 8.d odd 2 1
336.2.q.g 4 56.k odd 6 1
504.2.s.i 4 24.h odd 2 1
504.2.s.i 4 168.s odd 6 1
1008.2.s.r 4 24.f even 2 1
1008.2.s.r 4 168.v even 6 1
1176.2.a.k 2 56.p even 6 1
1176.2.a.n 2 56.j odd 6 1
1176.2.q.l 4 56.h odd 2 1
1176.2.q.l 4 56.j odd 6 1
1344.2.q.w 4 1.a even 1 1 trivial
1344.2.q.w 4 7.c even 3 1 inner
1344.2.q.x 4 4.b odd 2 1
1344.2.q.x 4 28.g odd 6 1
2352.2.a.ba 2 56.m even 6 1
2352.2.a.bf 2 56.k odd 6 1
2352.2.q.bf 4 56.e even 2 1
2352.2.q.bf 4 56.m even 6 1
3528.2.a.bd 2 168.ba even 6 1
3528.2.a.bk 2 168.s odd 6 1
3528.2.s.bk 4 168.i even 2 1
3528.2.s.bk 4 168.ba even 6 1
7056.2.a.ch 2 168.be odd 6 1
7056.2.a.cu 2 168.v even 6 1
9408.2.a.dj 2 7.d odd 6 1
9408.2.a.dp 2 28.g odd 6 1
9408.2.a.dw 2 28.f even 6 1
9408.2.a.ec 2 7.c even 3 1

Hecke kernels

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

\( T_{5}^{4} + T_{5}^{3} + 15T_{5}^{2} - 14T_{5} + 196 \) Copy content Toggle raw display
\( T_{11}^{4} + T_{11}^{3} + 15T_{11}^{2} - 14T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{2} + 5T_{13} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( (T^{2} + 3 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} + \cdots + 196 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T - 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 5 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 3 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 7 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 7 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{3} + \cdots + 196 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots + 1681 \) Copy content Toggle raw display
$83$ \( (T^{2} + 7 T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 6 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( (T^{2} - 25 T + 142)^{2} \) Copy content Toggle raw display
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