Properties

Label 264.1.p.a
Level $264$
Weight $1$
Character orbit 264.p
Self dual yes
Analytic conductor $0.132$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -8, -264, 33
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [264,1,Mod(131,264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(264, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("264.131");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 264.p (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: \(0.131753163335\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{33})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.2112.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} + q^{16} + 2 q^{17} - q^{18} - q^{22} + q^{24} - q^{25} - q^{27} - q^{32} - q^{33} - 2 q^{34} + q^{36} - 2 q^{41} + q^{44} - q^{48} - q^{49} + q^{50} - 2 q^{51} + q^{54} + q^{64} + q^{66} + 2 q^{67} + 2 q^{68} - q^{72} + q^{75} + q^{81} + 2 q^{82} - 2 q^{83} - q^{88} + q^{96} - 2 q^{97} + q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/264\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(133\) \(145\) \(199\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1
0
−1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
33.d even 2 1 RM by \(\Q(\sqrt{33}) \)
264.p odd 2 1 CM by \(\Q(\sqrt{-66}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 264.1.p.a 1
3.b odd 2 1 264.1.p.b yes 1
4.b odd 2 1 1056.1.p.a 1
8.b even 2 1 1056.1.p.a 1
8.d odd 2 1 CM 264.1.p.a 1
11.b odd 2 1 264.1.p.b yes 1
11.c even 5 4 2904.1.r.f 4
11.d odd 10 4 2904.1.r.b 4
12.b even 2 1 1056.1.p.b 1
24.f even 2 1 264.1.p.b yes 1
24.h odd 2 1 1056.1.p.b 1
33.d even 2 1 RM 264.1.p.a 1
33.f even 10 4 2904.1.r.f 4
33.h odd 10 4 2904.1.r.b 4
44.c even 2 1 1056.1.p.b 1
88.b odd 2 1 1056.1.p.b 1
88.g even 2 1 264.1.p.b yes 1
88.k even 10 4 2904.1.r.b 4
88.l odd 10 4 2904.1.r.f 4
132.d odd 2 1 1056.1.p.a 1
264.m even 2 1 1056.1.p.a 1
264.p odd 2 1 CM 264.1.p.a 1
264.r odd 10 4 2904.1.r.f 4
264.w even 10 4 2904.1.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.1.p.a 1 1.a even 1 1 trivial
264.1.p.a 1 8.d odd 2 1 CM
264.1.p.a 1 33.d even 2 1 RM
264.1.p.a 1 264.p odd 2 1 CM
264.1.p.b yes 1 3.b odd 2 1
264.1.p.b yes 1 11.b odd 2 1
264.1.p.b yes 1 24.f even 2 1
264.1.p.b yes 1 88.g even 2 1
1056.1.p.a 1 4.b odd 2 1
1056.1.p.a 1 8.b even 2 1
1056.1.p.a 1 132.d odd 2 1
1056.1.p.a 1 264.m even 2 1
1056.1.p.b 1 12.b even 2 1
1056.1.p.b 1 24.h odd 2 1
1056.1.p.b 1 44.c even 2 1
1056.1.p.b 1 88.b odd 2 1
2904.1.r.b 4 11.d odd 10 4
2904.1.r.b 4 33.h odd 10 4
2904.1.r.b 4 88.k even 10 4
2904.1.r.b 4 264.w even 10 4
2904.1.r.f 4 11.c even 5 4
2904.1.r.f 4 33.f even 10 4
2904.1.r.f 4 88.l odd 10 4
2904.1.r.f 4 264.r odd 10 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17} - 2 \) acting on \(S_{1}^{\mathrm{new}}(264, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) 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
$47$ \( T \) 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
$67$ \( T - 2 \) 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
$83$ \( T + 2 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
show more
show less