Properties

Label 1392.1.i.a
Level $1392$
Weight $1$
Character orbit 1392.i
Self dual yes
Analytic conductor $0.695$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -87
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1392,1,Mod(1217,1392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1392, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1392.1217");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1392 = 2^{4} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1392.i (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.87.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.484416.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{7} + q^{9} + q^{11} - q^{13} - q^{17} - q^{21} + q^{25} - q^{27} + q^{29} - q^{33} + q^{39} + 2 q^{41} + q^{47} + q^{51} + q^{63} + q^{67} - q^{75} + q^{77} + q^{81} - q^{87} - q^{89} - q^{91} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1392\mathbb{Z}\right)^\times\).

\(n\) \(175\) \(929\) \(1045\) \(1249\)
\(\chi(n)\) \(0\) \(1\) \(0\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
0
0 −1.00000 0 0 0 1.00000 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
87.d odd 2 1 CM by \(\Q(\sqrt{-87}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1392.1.i.a 1
3.b odd 2 1 1392.1.i.b 1
4.b odd 2 1 87.1.d.a 1
12.b even 2 1 87.1.d.b yes 1
20.d odd 2 1 2175.1.h.b 1
20.e even 4 2 2175.1.b.a 2
29.b even 2 1 1392.1.i.b 1
36.f odd 6 2 2349.1.h.b 2
36.h even 6 2 2349.1.h.a 2
60.h even 2 1 2175.1.h.a 1
60.l odd 4 2 2175.1.b.b 2
87.d odd 2 1 CM 1392.1.i.a 1
116.d odd 2 1 87.1.d.b yes 1
116.e even 4 2 2523.1.b.b 2
116.h odd 14 6 2523.1.h.a 6
116.j odd 14 6 2523.1.h.b 6
116.l even 28 12 2523.1.j.b 12
348.b even 2 1 87.1.d.a 1
348.k odd 4 2 2523.1.b.b 2
348.s even 14 6 2523.1.h.a 6
348.t even 14 6 2523.1.h.b 6
348.v odd 28 12 2523.1.j.b 12
580.e odd 2 1 2175.1.h.a 1
580.o even 4 2 2175.1.b.b 2
1044.o odd 6 2 2349.1.h.a 2
1044.t even 6 2 2349.1.h.b 2
1740.k even 2 1 2175.1.h.b 1
1740.v odd 4 2 2175.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 4.b odd 2 1
87.1.d.a 1 348.b even 2 1
87.1.d.b yes 1 12.b even 2 1
87.1.d.b yes 1 116.d odd 2 1
1392.1.i.a 1 1.a even 1 1 trivial
1392.1.i.a 1 87.d odd 2 1 CM
1392.1.i.b 1 3.b odd 2 1
1392.1.i.b 1 29.b even 2 1
2175.1.b.a 2 20.e even 4 2
2175.1.b.a 2 1740.v odd 4 2
2175.1.b.b 2 60.l odd 4 2
2175.1.b.b 2 580.o even 4 2
2175.1.h.a 1 60.h even 2 1
2175.1.h.a 1 580.e odd 2 1
2175.1.h.b 1 20.d odd 2 1
2175.1.h.b 1 1740.k even 2 1
2349.1.h.a 2 36.h even 6 2
2349.1.h.a 2 1044.o odd 6 2
2349.1.h.b 2 36.f odd 6 2
2349.1.h.b 2 1044.t even 6 2
2523.1.b.b 2 116.e even 4 2
2523.1.b.b 2 348.k odd 4 2
2523.1.h.a 6 116.h odd 14 6
2523.1.h.a 6 348.s even 14 6
2523.1.h.b 6 116.j odd 14 6
2523.1.h.b 6 348.t even 14 6
2523.1.j.b 12 116.l even 28 12
2523.1.j.b 12 348.v odd 28 12

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1392, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 1 \) Copy content Toggle raw display
$17$ \( T + 1 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 1 \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T - 1 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 1 \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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