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Properties

Label 585.2.bf.b
Level $585$
Weight $2$
Character orbit 585.bf
Analytic conductor $4.671$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(199,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bf (of order \(6\), degree \(2\), 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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{4} + 4 q^{10} + 16 q^{16} + 24 q^{19} + 8 q^{25} - 48 q^{40} - 48 q^{46} - 16 q^{49} + 28 q^{61} - 48 q^{64} - 144 q^{76} + 40 q^{79} + 12 q^{85} + 4 q^{91} - 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1 −1.12762 + 1.95310i 0 −1.54307 − 2.67267i −1.52448 − 1.63584i 0 1.53152 + 2.65268i 2.44949 0 4.91399 − 1.13285i
199.2 −1.12762 + 1.95310i 0 −1.54307 − 2.67267i −1.52448 + 1.63584i 0 −1.53152 − 2.65268i 2.44949 0 −1.47592 − 4.82206i
199.3 −0.779103 + 1.34945i 0 −0.214003 − 0.370665i 2.08241 − 0.814609i 0 −1.92183 − 3.32870i −2.44949 0 −0.523137 + 3.44476i
199.4 −0.779103 + 1.34945i 0 −0.214003 − 0.370665i 2.08241 + 0.814609i 0 1.92183 + 3.32870i −2.44949 0 −2.72168 + 2.17543i
199.5 −0.348519 + 0.603653i 0 0.757068 + 1.31128i −1.15739 − 1.91323i 0 −0.459373 − 0.795657i −2.44949 0 1.55830 − 0.0318667i
199.6 −0.348519 + 0.603653i 0 0.757068 + 1.31128i −1.15739 + 1.91323i 0 0.459373 + 0.795657i −2.44949 0 −0.751553 − 1.36546i
199.7 0.348519 − 0.603653i 0 0.757068 + 1.31128i 1.15739 − 1.91323i 0 0.459373 + 0.795657i 2.44949 0 −0.751553 − 1.36546i
199.8 0.348519 − 0.603653i 0 0.757068 + 1.31128i 1.15739 + 1.91323i 0 −0.459373 − 0.795657i 2.44949 0 1.55830 − 0.0318667i
199.9 0.779103 − 1.34945i 0 −0.214003 − 0.370665i −2.08241 − 0.814609i 0 1.92183 + 3.32870i 2.44949 0 −2.72168 + 2.17543i
199.10 0.779103 − 1.34945i 0 −0.214003 − 0.370665i −2.08241 + 0.814609i 0 −1.92183 − 3.32870i 2.44949 0 −0.523137 + 3.44476i
199.11 1.12762 − 1.95310i 0 −1.54307 − 2.67267i 1.52448 − 1.63584i 0 −1.53152 − 2.65268i −2.44949 0 −1.47592 − 4.82206i
199.12 1.12762 − 1.95310i 0 −1.54307 − 2.67267i 1.52448 + 1.63584i 0 1.53152 + 2.65268i −2.44949 0 4.91399 − 1.13285i
244.1 −1.12762 − 1.95310i 0 −1.54307 + 2.67267i −1.52448 − 1.63584i 0 −1.53152 + 2.65268i 2.44949 0 −1.47592 + 4.82206i
244.2 −1.12762 − 1.95310i 0 −1.54307 + 2.67267i −1.52448 + 1.63584i 0 1.53152 − 2.65268i 2.44949 0 4.91399 + 1.13285i
244.3 −0.779103 − 1.34945i 0 −0.214003 + 0.370665i 2.08241 − 0.814609i 0 1.92183 − 3.32870i −2.44949 0 −2.72168 − 2.17543i
244.4 −0.779103 − 1.34945i 0 −0.214003 + 0.370665i 2.08241 + 0.814609i 0 −1.92183 + 3.32870i −2.44949 0 −0.523137 − 3.44476i
244.5 −0.348519 − 0.603653i 0 0.757068 − 1.31128i −1.15739 − 1.91323i 0 0.459373 − 0.795657i −2.44949 0 −0.751553 + 1.36546i
244.6 −0.348519 − 0.603653i 0 0.757068 − 1.31128i −1.15739 + 1.91323i 0 −0.459373 + 0.795657i −2.44949 0 1.55830 + 0.0318667i
244.7 0.348519 + 0.603653i 0 0.757068 − 1.31128i 1.15739 − 1.91323i 0 −0.459373 + 0.795657i 2.44949 0 1.55830 + 0.0318667i
244.8 0.348519 + 0.603653i 0 0.757068 − 1.31128i 1.15739 + 1.91323i 0 0.459373 − 0.795657i 2.44949 0 −0.751553 + 1.36546i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.e even 6 1 inner
15.d odd 2 1 inner
39.h odd 6 1 inner
65.l even 6 1 inner
195.y odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bf.b ✓ 24
3.b odd 2 1 inner 585.2.bf.b ✓ 24
5.b even 2 1 inner 585.2.bf.b ✓ 24
13.e even 6 1 inner 585.2.bf.b ✓ 24
15.d odd 2 1 inner 585.2.bf.b ✓ 24
39.h odd 6 1 inner 585.2.bf.b ✓ 24
65.l even 6 1 inner 585.2.bf.b ✓ 24
195.y odd 6 1 inner 585.2.bf.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.bf.b ✓ 24 1.a even 1 1 trivial
585.2.bf.b ✓ 24 3.b odd 2 1 inner
585.2.bf.b ✓ 24 5.b even 2 1 inner
585.2.bf.b ✓ 24 13.e even 6 1 inner
585.2.bf.b ✓ 24 15.d odd 2 1 inner
585.2.bf.b ✓ 24 39.h odd 6 1 inner
585.2.bf.b ✓ 24 65.l even 6 1 inner
585.2.bf.b ✓ 24 195.y odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8T_{2}^{10} + 48T_{2}^{8} + 116T_{2}^{6} + 208T_{2}^{4} + 96T_{2}^{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display