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Properties

Label 728.2.bm.c
Level $728$
Weight $2$
Character orbit 728.bm
Analytic conductor $5.813$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [728,2,Mod(225,728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("728.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.bm (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: \(5.81310926715\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{3} - 18 q^{9} + 12 q^{11} + 8 q^{17} - 12 q^{19} + 2 q^{23} - 28 q^{25} - 20 q^{27} + 2 q^{29} - 18 q^{33} - 8 q^{35} + 60 q^{37} + 18 q^{39} - 6 q^{41} + 24 q^{43} - 72 q^{45} + 12 q^{49} - 72 q^{51}+ \cdots - 54 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1 0 −1.56651 + 2.71328i 0 − 2.28231i 0 0.866025 − 0.500000i 0 −3.40791 − 5.90267i 0
225.2 0 −1.45593 + 2.52174i 0 − 2.05353i 0 −0.866025 + 0.500000i 0 −2.73946 − 4.74488i 0
225.3 0 −1.21632 + 2.10672i 0 3.98804i 0 −0.866025 + 0.500000i 0 −1.45886 − 2.52682i 0
225.4 0 −0.735217 + 1.27343i 0 − 2.91465i 0 0.866025 − 0.500000i 0 0.418913 + 0.725579i 0
225.5 0 −0.509942 + 0.883245i 0 − 2.02001i 0 −0.866025 + 0.500000i 0 0.979919 + 1.69727i 0
225.6 0 −0.266360 + 0.461349i 0 3.61008i 0 0.866025 − 0.500000i 0 1.35810 + 2.35231i 0
225.7 0 −0.223973 + 0.387932i 0 − 0.513505i 0 0.866025 − 0.500000i 0 1.39967 + 2.42430i 0
225.8 0 0.0150934 − 0.0261426i 0 0.691852i 0 −0.866025 + 0.500000i 0 1.49954 + 2.59729i 0
225.9 0 0.879950 − 1.52412i 0 0.457188i 0 0.866025 − 0.500000i 0 −0.0486252 − 0.0842213i 0
225.10 0 1.01740 − 1.76219i 0 2.24692i 0 −0.866025 + 0.500000i 0 −0.570204 − 0.987622i 0
225.11 0 1.41211 − 2.44584i 0 − 4.08885i 0 0.866025 − 0.500000i 0 −2.48810 − 4.30952i 0
225.12 0 1.64970 − 2.85736i 0 − 0.585323i 0 −0.866025 + 0.500000i 0 −3.94299 − 6.82946i 0
673.1 0 −1.56651 − 2.71328i 0 2.28231i 0 0.866025 + 0.500000i 0 −3.40791 + 5.90267i 0
673.2 0 −1.45593 − 2.52174i 0 2.05353i 0 −0.866025 − 0.500000i 0 −2.73946 + 4.74488i 0
673.3 0 −1.21632 − 2.10672i 0 − 3.98804i 0 −0.866025 − 0.500000i 0 −1.45886 + 2.52682i 0
673.4 0 −0.735217 − 1.27343i 0 2.91465i 0 0.866025 + 0.500000i 0 0.418913 − 0.725579i 0
673.5 0 −0.509942 − 0.883245i 0 2.02001i 0 −0.866025 − 0.500000i 0 0.979919 − 1.69727i 0
673.6 0 −0.266360 − 0.461349i 0 − 3.61008i 0 0.866025 + 0.500000i 0 1.35810 − 2.35231i 0
673.7 0 −0.223973 − 0.387932i 0 0.513505i 0 0.866025 + 0.500000i 0 1.39967 − 2.42430i 0
673.8 0 0.0150934 + 0.0261426i 0 − 0.691852i 0 −0.866025 − 0.500000i 0 1.49954 − 2.59729i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 728.2.bm.c ✓ 24
4.b odd 2 1 1456.2.cc.g 24
13.e even 6 1 inner 728.2.bm.c ✓ 24
13.f odd 12 1 9464.2.a.bl 12
13.f odd 12 1 9464.2.a.bm 12
52.i odd 6 1 1456.2.cc.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.bm.c ✓ 24 1.a even 1 1 trivial
728.2.bm.c ✓ 24 13.e even 6 1 inner
1456.2.cc.g 24 4.b odd 2 1
1456.2.cc.g 24 52.i odd 6 1
9464.2.a.bl 12 13.f odd 12 1
9464.2.a.bm 12 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 2 T_{3}^{23} + 29 T_{3}^{22} + 54 T_{3}^{21} + 521 T_{3}^{20} + 932 T_{3}^{19} + 5754 T_{3}^{18} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\). Copy content Toggle raw display