Properties

Label 1156.2.a.c
Level $1156$
Weight $2$
Character orbit 1156.a
Self dual yes
Analytic conductor $9.231$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,2,Mod(1,1156)]
 
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([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

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: \(9.23070647366\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta q^{3} + 2 \beta q^{5} + 3 \beta q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} + 2 \beta q^{5} + 3 \beta q^{7} - q^{9} + \beta q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 6 q^{21} + \beta q^{23} + 3 q^{25} - 4 \beta q^{27} + 2 \beta q^{29} + 3 \beta q^{31} + 2 q^{33} + 12 q^{35} - 6 \beta q^{37} - 4 \beta q^{39} - 8 \beta q^{41} - 8 q^{43} - 2 \beta q^{45} - 12 q^{47} + 11 q^{49} + 6 q^{53} + 4 q^{55} + 4 \beta q^{57} + 6 \beta q^{61} - 3 \beta q^{63} - 8 \beta q^{65} - 4 q^{67} + 2 q^{69} + 5 \beta q^{71} + 3 \beta q^{75} + 6 q^{77} - 3 \beta q^{79} - 5 q^{81} + 4 q^{87} + 12 q^{89} - 12 \beta q^{91} + 6 q^{93} + 8 \beta q^{95} - \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} - 8 q^{13} + 8 q^{15} + 8 q^{19} + 12 q^{21} + 6 q^{25} + 4 q^{33} + 24 q^{35} - 16 q^{43} - 24 q^{47} + 22 q^{49} + 12 q^{53} + 8 q^{55} - 8 q^{67} + 4 q^{69} + 12 q^{77} - 10 q^{81} + 8 q^{87} + 24 q^{89} + 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.41421 0 −2.82843 0 −4.24264 0 −1.00000 0
1.2 0 1.41421 0 2.82843 0 4.24264 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.a.c 2
4.b odd 2 1 4624.2.a.n 2
17.b even 2 1 inner 1156.2.a.c 2
17.c even 4 2 68.2.b.a 2
17.d even 8 2 1156.2.e.a 2
17.d even 8 2 1156.2.e.b 2
17.e odd 16 8 1156.2.h.d 8
51.f odd 4 2 612.2.b.a 2
68.d odd 2 1 4624.2.a.n 2
68.f odd 4 2 272.2.b.c 2
85.f odd 4 2 1700.2.g.a 4
85.i odd 4 2 1700.2.g.a 4
85.j even 4 2 1700.2.c.a 2
119.f odd 4 2 3332.2.b.a 2
136.i even 4 2 1088.2.b.e 2
136.j odd 4 2 1088.2.b.f 2
204.l even 4 2 2448.2.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 17.c even 4 2
272.2.b.c 2 68.f odd 4 2
612.2.b.a 2 51.f odd 4 2
1088.2.b.e 2 136.i even 4 2
1088.2.b.f 2 136.j odd 4 2
1156.2.a.c 2 1.a even 1 1 trivial
1156.2.a.c 2 17.b even 2 1 inner
1156.2.e.a 2 17.d even 8 2
1156.2.e.b 2 17.d even 8 2
1156.2.h.d 8 17.e odd 16 8
1700.2.c.a 2 85.j even 4 2
1700.2.g.a 4 85.f odd 4 2
1700.2.g.a 4 85.i odd 4 2
2448.2.c.d 2 204.l even 4 2
3332.2.b.a 2 119.f odd 4 2
4624.2.a.n 2 4.b odd 2 1
4624.2.a.n 2 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 8 \) Copy content Toggle raw display
$7$ \( T^{2} - 18 \) Copy content Toggle raw display
$11$ \( T^{2} - 2 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2 \) Copy content Toggle raw display
$29$ \( T^{2} - 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 18 \) Copy content Toggle raw display
$37$ \( T^{2} - 72 \) Copy content Toggle raw display
$41$ \( T^{2} - 128 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 72 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 50 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 18 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
show more
show less