↑

Properties

Label 3392.1.bn.a
Level $3392$
Weight $1$
Character orbit 3392.bn
Analytic conductor $1.693$
Analytic rank $0$
Dimension $12$
Projective image $D_{26}$
CM discriminant -4
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3392,1,Mod(255,3392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3392, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([13, 0, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3392.255");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3392 = 2^{6} \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3392.bn (of order \(26\), degree \(12\), 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: \(1.69282852285\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 848)
Projective image: \(D_{26}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{26} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{26}^{9} - \zeta_{26}^{2}) q^{5} - \zeta_{26}^{12} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{26}^{9} - \zeta_{26}^{2}) q^{5} - \zeta_{26}^{12} q^{9} + ( - \zeta_{26}^{6} + \zeta_{26}) q^{13} + (\zeta_{26}^{10} - \zeta_{26}^{7}) q^{17} + (\zeta_{26}^{11} + \cdots + \zeta_{26}^{4}) q^{25} + \cdots + (\zeta_{26}^{11} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{9} + 2 q^{13} - 2 q^{17} - q^{25} - 2 q^{29} - 2 q^{37} - q^{49} - q^{53} - q^{81} - 11 q^{89} - 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3392\mathbb{Z}\right)^\times\).

\(n\) \(319\) \(1857\) \(2757\)
\(\chi(n)\) \(-1\) \(\zeta_{26}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.354605 − 0.935016i
−0.120537 + 0.992709i
−0.885456 − 0.464723i
−0.885456 + 0.464723i
0.970942 − 0.239316i
−0.568065 − 0.822984i
0.748511 − 0.663123i
0.970942 + 0.239316i
−0.568065 + 0.822984i
0.354605 + 0.935016i
0.748511 + 0.663123i
−0.120537 − 0.992709i
0 0 0 0.869047 − 0.329586i 0 0 0 0.354605 + 0.935016i 0
767.1 0 0 0 1.85640 − 0.225408i 0 0 0 −0.120537 − 0.992709i 0
1151.1 0 0 0 −0.922670 − 1.75800i 0 0 0 −0.885456 + 0.464723i 0
1279.1 0 0 0 −0.922670 + 1.75800i 0 0 0 −0.885456 − 0.464723i 0
1407.1 0 0 0 −0.317391 + 1.28771i 0 0 0 0.970942 + 0.239316i 0
1471.1 0 0 0 −0.393906 − 0.271894i 0 0 0 −0.568065 + 0.822984i 0
1599.1 0 0 0 −1.09148 + 1.23202i 0 0 0 0.748511 + 0.663123i 0
1919.1 0 0 0 −0.317391 − 1.28771i 0 0 0 0.970942 − 0.239316i 0
2495.1 0 0 0 −0.393906 + 0.271894i 0 0 0 −0.568065 − 0.822984i 0
2687.1 0 0 0 0.869047 + 0.329586i 0 0 0 0.354605 − 0.935016i 0
2815.1 0 0 0 −1.09148 − 1.23202i 0 0 0 0.748511 − 0.663123i 0
2879.1 0 0 0 1.85640 + 0.225408i 0 0 0 −0.120537 + 0.992709i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
53.e even 26 1 inner
212.h odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3392.1.bn.a 12
4.b odd 2 1 CM 3392.1.bn.a 12
8.b even 2 1 848.1.x.a ✓ 12
8.d odd 2 1 848.1.x.a ✓ 12
53.e even 26 1 inner 3392.1.bn.a 12
212.h odd 26 1 inner 3392.1.bn.a 12
424.n even 26 1 848.1.x.a ✓ 12
424.q odd 26 1 848.1.x.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
848.1.x.a ✓ 12 8.b even 2 1
848.1.x.a ✓ 12 8.d odd 2 1
848.1.x.a ✓ 12 424.n even 26 1
848.1.x.a ✓ 12 424.q odd 26 1
3392.1.bn.a 12 1.a even 1 1 trivial
3392.1.bn.a 12 4.b odd 2 1 CM
3392.1.bn.a 12 53.e even 26 1 inner
3392.1.bn.a 12 212.h odd 26 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3392, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 13 T^{9} + \cdots + 13 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} - 13 T^{7} + \cdots + 13 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} + 13 T^{7} + \cdots + 13 \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} + 13 T^{9} + \cdots + 13 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} + 11 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} + 11 T^{11} + \cdots + 1 \) Copy content Toggle raw display
show more
show less