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Properties

Label 528.2.bl.a
Level $528$
Weight $2$
Character orbit 528.bl
Analytic conductor $4.216$
Analytic rank $0$
Dimension $32$
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] = [528,2,Mod(47,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 5, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.bl (of order \(10\), degree \(4\), 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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 18 q^{9} + 8 q^{13} - 4 q^{21} + 24 q^{25} + 62 q^{33} + 44 q^{45} + 80 q^{49} - 30 q^{57} + 16 q^{61} - 64 q^{69} + 4 q^{73} - 54 q^{81} - 112 q^{85} - 58 q^{93} - 104 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.73203 − 0.00912796i 0 2.33144 − 3.20895i 0 3.80774 + 1.23721i 0 2.99983 + 0.0316197i 0
47.2 0 −1.54571 − 0.781516i 0 −0.303754 + 0.418082i 0 −2.73941 − 0.890090i 0 1.77847 + 2.41600i 0
47.3 0 −1.39587 + 1.02544i 0 −2.33144 + 3.20895i 0 3.80774 + 1.23721i 0 0.896927 − 2.86278i 0
47.4 0 −0.791146 + 1.54081i 0 0.303754 − 0.418082i 0 −2.73941 − 0.890090i 0 −1.74818 − 2.43801i 0
47.5 0 0.791146 − 1.54081i 0 0.303754 − 0.418082i 0 2.73941 + 0.890090i 0 −1.74818 − 2.43801i 0
47.6 0 1.39587 − 1.02544i 0 −2.33144 + 3.20895i 0 −3.80774 − 1.23721i 0 0.896927 − 2.86278i 0
47.7 0 1.54571 + 0.781516i 0 −0.303754 + 0.418082i 0 2.73941 + 0.890090i 0 1.77847 + 2.41600i 0
47.8 0 1.73203 + 0.00912796i 0 2.33144 − 3.20895i 0 −3.80774 − 1.23721i 0 2.99983 + 0.0316197i 0
191.1 0 −1.73203 + 0.00912796i 0 2.33144 + 3.20895i 0 3.80774 − 1.23721i 0 2.99983 − 0.0316197i 0
191.2 0 −1.54571 + 0.781516i 0 −0.303754 − 0.418082i 0 −2.73941 + 0.890090i 0 1.77847 − 2.41600i 0
191.3 0 −1.39587 − 1.02544i 0 −2.33144 − 3.20895i 0 3.80774 − 1.23721i 0 0.896927 + 2.86278i 0
191.4 0 −0.791146 − 1.54081i 0 0.303754 + 0.418082i 0 −2.73941 + 0.890090i 0 −1.74818 + 2.43801i 0
191.5 0 0.791146 + 1.54081i 0 0.303754 + 0.418082i 0 2.73941 − 0.890090i 0 −1.74818 + 2.43801i 0
191.6 0 1.39587 + 1.02544i 0 −2.33144 − 3.20895i 0 −3.80774 + 1.23721i 0 0.896927 + 2.86278i 0
191.7 0 1.54571 − 0.781516i 0 −0.303754 − 0.418082i 0 2.73941 − 0.890090i 0 1.77847 − 2.41600i 0
191.8 0 1.73203 − 0.00912796i 0 2.33144 + 3.20895i 0 −3.80774 + 1.23721i 0 2.99983 − 0.0316197i 0
335.1 0 −1.53878 − 0.795085i 0 −3.55001 − 1.15347i 0 −1.58848 − 2.18636i 0 1.73568 + 2.44692i 0
335.2 0 −1.51340 + 0.842387i 0 1.36731 + 0.444265i 0 −0.688069 − 0.947045i 0 1.58077 − 2.54974i 0
335.3 0 −1.23168 − 1.21777i 0 3.55001 + 1.15347i 0 1.58848 + 2.18636i 0 0.0340680 + 2.99981i 0
335.4 0 −0.333491 + 1.69964i 0 −1.36731 − 0.444265i 0 −0.688069 − 0.947045i 0 −2.77757 − 1.13363i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
11.c even 5 1 inner
12.b even 2 1 inner
33.h odd 10 1 inner
44.h odd 10 1 inner
132.o even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.bl.a ✓ 32
3.b odd 2 1 inner 528.2.bl.a ✓ 32
4.b odd 2 1 inner 528.2.bl.a ✓ 32
11.c even 5 1 inner 528.2.bl.a ✓ 32
12.b even 2 1 inner 528.2.bl.a ✓ 32
33.h odd 10 1 inner 528.2.bl.a ✓ 32
44.h odd 10 1 inner 528.2.bl.a ✓ 32
132.o even 10 1 inner 528.2.bl.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.bl.a ✓ 32 1.a even 1 1 trivial
528.2.bl.a ✓ 32 3.b odd 2 1 inner
528.2.bl.a ✓ 32 4.b odd 2 1 inner
528.2.bl.a ✓ 32 11.c even 5 1 inner
528.2.bl.a ✓ 32 12.b even 2 1 inner
528.2.bl.a ✓ 32 33.h odd 10 1 inner
528.2.bl.a ✓ 32 44.h odd 10 1 inner
528.2.bl.a ✓ 32 132.o even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} - 16 T_{5}^{14} + 267 T_{5}^{12} - 4448 T_{5}^{10} + 60630 T_{5}^{8} - 166672 T_{5}^{6} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display