Properties

Label 1156.1.c.b
Level $1156$
Weight $1$
Character orbit 1156.c
Self dual yes
Analytic conductor $0.577$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
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] = [1156,1,Mod(579,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.579");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.c (of order \(2\), degree \(1\), 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: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $D_8$
Artin field: Galois closure of 8.2.26261675072.1

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta q^{5} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta q^{5} - q^{8} + q^{9} + \beta q^{10} + q^{16} - q^{18} - \beta q^{20} + q^{25} + \beta q^{29} - q^{32} + q^{36} + \beta q^{37} + \beta q^{40} + \beta q^{41} - \beta q^{45} + q^{49} - q^{50} - \beta q^{58} - \beta q^{61} + q^{64} - q^{72} + \beta q^{73} - \beta q^{74} - \beta q^{80} + q^{81} - \beta q^{82} + \beta q^{90} - \beta q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{9} + 2 q^{16} - 2 q^{18} + 2 q^{25} - 2 q^{32} + 2 q^{36} + 2 q^{49} - 2 q^{50} + 2 q^{64} - 2 q^{72} + 2 q^{81} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
579.1
1.41421
−1.41421
−1.00000 0 1.00000 −1.41421 0 0 −1.00000 1.00000 1.41421
579.2 −1.00000 0 1.00000 1.41421 0 0 −1.00000 1.00000 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 inner
68.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.c.b 2
4.b odd 2 1 CM 1156.1.c.b 2
17.b even 2 1 inner 1156.1.c.b 2
17.c even 4 2 1156.1.d.a 2
17.d even 8 2 68.1.f.a 2
17.d even 8 2 1156.1.f.b 2
17.e odd 16 8 1156.1.g.b 8
51.g odd 8 2 612.1.l.a 2
68.d odd 2 1 inner 1156.1.c.b 2
68.f odd 4 2 1156.1.d.a 2
68.g odd 8 2 68.1.f.a 2
68.g odd 8 2 1156.1.f.b 2
68.i even 16 8 1156.1.g.b 8
85.k odd 8 2 1700.1.n.b 2
85.m even 8 2 1700.1.p.a 2
85.n odd 8 2 1700.1.n.a 2
119.l odd 8 2 3332.1.m.b 2
119.q even 24 4 3332.1.bc.c 4
119.r odd 24 4 3332.1.bc.b 4
136.o even 8 2 1088.1.p.a 2
136.p odd 8 2 1088.1.p.a 2
204.p even 8 2 612.1.l.a 2
340.w even 8 2 1700.1.n.a 2
340.z even 8 2 1700.1.n.b 2
340.ba odd 8 2 1700.1.p.a 2
476.w even 8 2 3332.1.m.b 2
476.bg odd 24 4 3332.1.bc.c 4
476.bj even 24 4 3332.1.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 17.d even 8 2
68.1.f.a 2 68.g odd 8 2
612.1.l.a 2 51.g odd 8 2
612.1.l.a 2 204.p even 8 2
1088.1.p.a 2 136.o even 8 2
1088.1.p.a 2 136.p odd 8 2
1156.1.c.b 2 1.a even 1 1 trivial
1156.1.c.b 2 4.b odd 2 1 CM
1156.1.c.b 2 17.b even 2 1 inner
1156.1.c.b 2 68.d odd 2 1 inner
1156.1.d.a 2 17.c even 4 2
1156.1.d.a 2 68.f odd 4 2
1156.1.f.b 2 17.d even 8 2
1156.1.f.b 2 68.g odd 8 2
1156.1.g.b 8 17.e odd 16 8
1156.1.g.b 8 68.i even 16 8
1700.1.n.a 2 85.n odd 8 2
1700.1.n.a 2 340.w even 8 2
1700.1.n.b 2 85.k odd 8 2
1700.1.n.b 2 340.z even 8 2
1700.1.p.a 2 85.m even 8 2
1700.1.p.a 2 340.ba odd 8 2
3332.1.m.b 2 119.l odd 8 2
3332.1.m.b 2 476.w even 8 2
3332.1.bc.b 4 119.r odd 24 4
3332.1.bc.b 4 476.bj even 24 4
3332.1.bc.c 4 119.q even 24 4
3332.1.bc.c 4 476.bg odd 24 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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