Properties

Label 1344.2.bl.h
Level $1344$
Weight $2$
Character orbit 1344.bl
Analytic conductor $10.732$
Analytic rank $0$
Dimension $2$
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(703,1344)]
 
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, 0, 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("1344.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.bl (of order \(6\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 4) q^{5} + ( - 3 \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 4) q^{5} + ( - 3 \zeta_{6} + 2) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + ( - 6 \zeta_{6} + 3) q^{13} + ( - 4 \zeta_{6} + 2) q^{15} + ( - 4 \zeta_{6} - 4) q^{17} + 7 \zeta_{6} q^{19} + ( - 2 \zeta_{6} - 1) q^{21} + ( - 7 \zeta_{6} + 7) q^{25} - q^{27} + (5 \zeta_{6} - 5) q^{31} + ( - 2 \zeta_{6} + 4) q^{33} + ( - 10 \zeta_{6} + 2) q^{35} + \zeta_{6} q^{37} + ( - 3 \zeta_{6} - 3) q^{39} + (12 \zeta_{6} - 6) q^{41} + ( - 2 \zeta_{6} + 1) q^{43} + ( - 2 \zeta_{6} - 2) q^{45} + 6 \zeta_{6} q^{47} + ( - 3 \zeta_{6} - 5) q^{49} + (4 \zeta_{6} - 8) q^{51} + 12 q^{55} + 7 q^{57} + (\zeta_{6} - 3) q^{63} - 18 \zeta_{6} q^{65} + ( - \zeta_{6} - 1) q^{67} + ( - 4 \zeta_{6} + 2) q^{71} + (5 \zeta_{6} + 5) q^{73} - 7 \zeta_{6} q^{75} + ( - 8 \zeta_{6} + 10) q^{77} + (9 \zeta_{6} - 18) q^{79} + (\zeta_{6} - 1) q^{81} + 6 q^{83} - 24 q^{85} + ( - 4 \zeta_{6} + 8) q^{89} + ( - 3 \zeta_{6} - 12) q^{91} + 5 \zeta_{6} q^{93} + (14 \zeta_{6} + 14) q^{95} + ( - 8 \zeta_{6} + 4) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{5} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{5} + q^{7} - q^{9} + 6 q^{11} - 12 q^{17} + 7 q^{19} - 4 q^{21} + 7 q^{25} - 2 q^{27} - 5 q^{31} + 6 q^{33} - 6 q^{35} + q^{37} - 9 q^{39} - 6 q^{45} + 6 q^{47} - 13 q^{49} - 12 q^{51} + 24 q^{55} + 14 q^{57} - 5 q^{63} - 18 q^{65} - 3 q^{67} + 15 q^{73} - 7 q^{75} + 12 q^{77} - 27 q^{79} - q^{81} + 12 q^{83} - 48 q^{85} + 12 q^{89} - 27 q^{91} + 5 q^{93} + 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 3.00000 1.73205i 0 0.500000 2.59808i 0 −0.500000 0.866025i 0
1279.1 0 0.500000 + 0.866025i 0 3.00000 + 1.73205i 0 0.500000 + 2.59808i 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
28.f 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.bl.h 2
4.b odd 2 1 1344.2.bl.d 2
7.d odd 6 1 1344.2.bl.d 2
8.b even 2 1 336.2.bl.a 2
8.d odd 2 1 336.2.bl.e yes 2
24.f even 2 1 1008.2.cs.m 2
24.h odd 2 1 1008.2.cs.n 2
28.f even 6 1 inner 1344.2.bl.h 2
56.e even 2 1 2352.2.bl.f 2
56.h odd 2 1 2352.2.bl.l 2
56.j odd 6 1 336.2.bl.e yes 2
56.j odd 6 1 2352.2.b.a 2
56.k odd 6 1 2352.2.b.a 2
56.k odd 6 1 2352.2.bl.l 2
56.m even 6 1 336.2.bl.a 2
56.m even 6 1 2352.2.b.h 2
56.p even 6 1 2352.2.b.h 2
56.p even 6 1 2352.2.bl.f 2
168.s odd 6 1 7056.2.b.l 2
168.v even 6 1 7056.2.b.a 2
168.ba even 6 1 1008.2.cs.m 2
168.ba even 6 1 7056.2.b.a 2
168.be odd 6 1 1008.2.cs.n 2
168.be odd 6 1 7056.2.b.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bl.a 2 8.b even 2 1
336.2.bl.a 2 56.m even 6 1
336.2.bl.e yes 2 8.d odd 2 1
336.2.bl.e yes 2 56.j odd 6 1
1008.2.cs.m 2 24.f even 2 1
1008.2.cs.m 2 168.ba even 6 1
1008.2.cs.n 2 24.h odd 2 1
1008.2.cs.n 2 168.be odd 6 1
1344.2.bl.d 2 4.b odd 2 1
1344.2.bl.d 2 7.d odd 6 1
1344.2.bl.h 2 1.a even 1 1 trivial
1344.2.bl.h 2 28.f even 6 1 inner
2352.2.b.a 2 56.j odd 6 1
2352.2.b.a 2 56.k odd 6 1
2352.2.b.h 2 56.m even 6 1
2352.2.b.h 2 56.p even 6 1
2352.2.bl.f 2 56.e even 2 1
2352.2.bl.f 2 56.p even 6 1
2352.2.bl.l 2 56.h odd 2 1
2352.2.bl.l 2 56.k odd 6 1
7056.2.b.a 2 168.v even 6 1
7056.2.b.a 2 168.ba even 6 1
7056.2.b.l 2 168.s odd 6 1
7056.2.b.l 2 168.be odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 6T_{5} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 27 \) Copy content Toggle raw display
$17$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 108 \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$79$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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