Properties

Label 1344.1.bn.a
Level $1344$
Weight $1$
Character orbit 1344.bn
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,1,Mod(65,1344)]
 
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, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.65");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1344.bn (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: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
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 84)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.588.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} - \zeta_{6} q^{9} + q^{13} + \zeta_{6} q^{19} - q^{21} + \zeta_{6}^{2} q^{25} + q^{27} + \zeta_{6}^{2} q^{31} - \zeta_{6} q^{37} + \zeta_{6}^{2} q^{39} - q^{43} + \zeta_{6}^{2} q^{49} - q^{57} + \zeta_{6} q^{61} - \zeta_{6}^{2} q^{63} - \zeta_{6}^{2} q^{67} - \zeta_{6}^{2} q^{73} - \zeta_{6} q^{75} - \zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{91} - \zeta_{6} q^{93} + q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{7} - q^{9} + 2 q^{13} + q^{19} - 2 q^{21} - q^{25} + 2 q^{27} - q^{31} - q^{37} - q^{39} - 2 q^{43} - q^{49} - 2 q^{57} + 2 q^{61} + q^{63} + q^{67} + q^{73} - q^{75} - q^{79} - q^{81} + q^{91} - q^{93} + 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 0 0 0.500000 0.866025i 0 −0.500000 + 0.866025i 0
641.1 0 −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.1.bn.a 2
3.b odd 2 1 CM 1344.1.bn.a 2
4.b odd 2 1 1344.1.bn.b 2
7.c even 3 1 inner 1344.1.bn.a 2
8.b even 2 1 336.1.bn.a 2
8.d odd 2 1 84.1.p.a 2
12.b even 2 1 1344.1.bn.b 2
21.h odd 6 1 inner 1344.1.bn.a 2
24.f even 2 1 84.1.p.a 2
24.h odd 2 1 336.1.bn.a 2
28.g odd 6 1 1344.1.bn.b 2
40.e odd 2 1 2100.1.bn.c 2
40.k even 4 2 2100.1.bh.a 4
56.e even 2 1 588.1.p.a 2
56.h odd 2 1 2352.1.bn.a 2
56.j odd 6 1 2352.1.d.b 1
56.j odd 6 1 2352.1.bn.a 2
56.k odd 6 1 84.1.p.a 2
56.k odd 6 1 588.1.c.b 1
56.m even 6 1 588.1.c.a 1
56.m even 6 1 588.1.p.a 2
56.p even 6 1 336.1.bn.a 2
56.p even 6 1 2352.1.d.a 1
72.l even 6 1 2268.1.m.a 2
72.l even 6 1 2268.1.bh.b 2
72.p odd 6 1 2268.1.m.a 2
72.p odd 6 1 2268.1.bh.b 2
84.n even 6 1 1344.1.bn.b 2
120.m even 2 1 2100.1.bn.c 2
120.q odd 4 2 2100.1.bh.a 4
168.e odd 2 1 588.1.p.a 2
168.i even 2 1 2352.1.bn.a 2
168.s odd 6 1 336.1.bn.a 2
168.s odd 6 1 2352.1.d.a 1
168.v even 6 1 84.1.p.a 2
168.v even 6 1 588.1.c.b 1
168.ba even 6 1 2352.1.d.b 1
168.ba even 6 1 2352.1.bn.a 2
168.be odd 6 1 588.1.c.a 1
168.be odd 6 1 588.1.p.a 2
280.bi odd 6 1 2100.1.bn.c 2
280.br even 12 2 2100.1.bh.a 4
504.ba odd 6 1 2268.1.bh.b 2
504.bt even 6 1 2268.1.m.a 2
504.ce odd 6 1 2268.1.m.a 2
504.cy even 6 1 2268.1.bh.b 2
840.cv even 6 1 2100.1.bn.c 2
840.dp odd 12 2 2100.1.bh.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 8.d odd 2 1
84.1.p.a 2 24.f even 2 1
84.1.p.a 2 56.k odd 6 1
84.1.p.a 2 168.v even 6 1
336.1.bn.a 2 8.b even 2 1
336.1.bn.a 2 24.h odd 2 1
336.1.bn.a 2 56.p even 6 1
336.1.bn.a 2 168.s odd 6 1
588.1.c.a 1 56.m even 6 1
588.1.c.a 1 168.be odd 6 1
588.1.c.b 1 56.k odd 6 1
588.1.c.b 1 168.v even 6 1
588.1.p.a 2 56.e even 2 1
588.1.p.a 2 56.m even 6 1
588.1.p.a 2 168.e odd 2 1
588.1.p.a 2 168.be odd 6 1
1344.1.bn.a 2 1.a even 1 1 trivial
1344.1.bn.a 2 3.b odd 2 1 CM
1344.1.bn.a 2 7.c even 3 1 inner
1344.1.bn.a 2 21.h odd 6 1 inner
1344.1.bn.b 2 4.b odd 2 1
1344.1.bn.b 2 12.b even 2 1
1344.1.bn.b 2 28.g odd 6 1
1344.1.bn.b 2 84.n even 6 1
2100.1.bh.a 4 40.k even 4 2
2100.1.bh.a 4 120.q odd 4 2
2100.1.bh.a 4 280.br even 12 2
2100.1.bh.a 4 840.dp odd 12 2
2100.1.bn.c 2 40.e odd 2 1
2100.1.bn.c 2 120.m even 2 1
2100.1.bn.c 2 280.bi odd 6 1
2100.1.bn.c 2 840.cv even 6 1
2268.1.m.a 2 72.l even 6 1
2268.1.m.a 2 72.p odd 6 1
2268.1.m.a 2 504.bt even 6 1
2268.1.m.a 2 504.ce odd 6 1
2268.1.bh.b 2 72.l even 6 1
2268.1.bh.b 2 72.p odd 6 1
2268.1.bh.b 2 504.ba odd 6 1
2268.1.bh.b 2 504.cy even 6 1
2352.1.d.a 1 56.p even 6 1
2352.1.d.a 1 168.s odd 6 1
2352.1.d.b 1 56.j odd 6 1
2352.1.d.b 1 168.ba even 6 1
2352.1.bn.a 2 56.h odd 2 1
2352.1.bn.a 2 56.j odd 6 1
2352.1.bn.a 2 168.i even 2 1
2352.1.bn.a 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{2} - T_{19} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1344, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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