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Properties

Label 756.4.x.a
Level $756$
Weight $4$
Character orbit 756.x
Analytic conductor $44.605$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,4,Mod(125,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.125");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.x (of order \(6\), degree \(2\), not 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: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 252)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 0 0 0 −10.5571 + 18.2854i 0 17.6959 + 5.46406i 0 0 0
125.2 0 0 0 −9.12012 + 15.7965i 0 −9.48165 − 15.9091i 0 0 0
125.3 0 0 0 −8.29874 + 14.3738i 0 18.2297 + 3.26749i 0 0 0
125.4 0 0 0 −7.82452 + 13.5525i 0 −13.7444 + 12.4134i 0 0 0
125.5 0 0 0 −6.03570 + 10.4541i 0 2.10370 − 18.4004i 0 0 0
125.6 0 0 0 −5.49690 + 9.52092i 0 6.85688 − 17.2042i 0 0 0
125.7 0 0 0 −5.16485 + 8.94579i 0 −17.2289 + 6.79448i 0 0 0
125.8 0 0 0 −3.53447 + 6.12188i 0 10.8305 + 15.0234i 0 0 0
125.9 0 0 0 −2.99997 + 5.19610i 0 −0.375477 + 18.5165i 0 0 0
125.10 0 0 0 −2.34269 + 4.05766i 0 −10.9369 + 14.9461i 0 0 0
125.11 0 0 0 −2.20656 + 3.82187i 0 −17.7037 + 5.43851i 0 0 0
125.12 0 0 0 −0.330097 + 0.571745i 0 −15.6440 − 9.91296i 0 0 0
125.13 0 0 0 0.330097 − 0.571745i 0 −0.762896 − 18.5045i 0 0 0
125.14 0 0 0 2.20656 − 3.82187i 0 13.5618 − 12.6126i 0 0 0
125.15 0 0 0 2.34269 − 4.05766i 0 18.4121 − 1.99859i 0 0 0
125.16 0 0 0 2.99997 − 5.19610i 0 16.2235 + 8.93305i 0 0 0
125.17 0 0 0 3.53447 − 6.12188i 0 7.59538 + 16.8911i 0 0 0
125.18 0 0 0 5.16485 − 8.94579i 0 14.4986 − 11.5234i 0 0 0
125.19 0 0 0 5.49690 − 9.52092i 0 −18.3277 − 2.66385i 0 0 0
125.20 0 0 0 6.03570 − 10.4541i 0 −16.9871 − 7.37834i 0 0 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.4.x.a 48
3.b odd 2 1 252.4.x.a ✓ 48
7.b odd 2 1 inner 756.4.x.a 48
9.c even 3 1 252.4.x.a ✓ 48
9.c even 3 1 2268.4.f.a 48
9.d odd 6 1 inner 756.4.x.a 48
9.d odd 6 1 2268.4.f.a 48
21.c even 2 1 252.4.x.a ✓ 48
63.l odd 6 1 252.4.x.a ✓ 48
63.l odd 6 1 2268.4.f.a 48
63.o even 6 1 inner 756.4.x.a 48
63.o even 6 1 2268.4.f.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.x.a ✓ 48 3.b odd 2 1
252.4.x.a ✓ 48 9.c even 3 1
252.4.x.a ✓ 48 21.c even 2 1
252.4.x.a ✓ 48 63.l odd 6 1
756.4.x.a 48 1.a even 1 1 trivial
756.4.x.a 48 7.b odd 2 1 inner
756.4.x.a 48 9.d odd 6 1 inner
756.4.x.a 48 63.o even 6 1 inner
2268.4.f.a 48 9.c even 3 1
2268.4.f.a 48 9.d odd 6 1
2268.4.f.a 48 63.l odd 6 1
2268.4.f.a 48 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(756, [\chi])\).