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Properties

Label 6728.2.a.bf
Level $6728$
Weight $2$
Character orbit 6728.a
Self dual yes
Analytic conductor $53.723$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6728,2,Mod(1,6728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6728 = 2^{3} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6728.a (trivial)

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: \(53.7233504799\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 232)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} + 8 q^{5} - 2 q^{7} + 32 q^{9} + 6 q^{11} + 2 q^{13} + 32 q^{15} + 20 q^{17} + 44 q^{19} + 4 q^{21} + 2 q^{23} + 32 q^{25} + 34 q^{27} + 10 q^{31} - 12 q^{33} - 32 q^{35} + 40 q^{37} + 18 q^{39} + 8 q^{41} + 30 q^{43} + 16 q^{45} + 34 q^{47} + 26 q^{49} - 28 q^{51} + 18 q^{53} + 50 q^{55} + 6 q^{57} - 64 q^{59} + 6 q^{61} - 28 q^{63} + 14 q^{65} - 40 q^{67} + 38 q^{69} + 52 q^{73} + 66 q^{75} + 90 q^{77} + 4 q^{79} + 40 q^{81} - 44 q^{83} + 40 q^{85} - 40 q^{89} - 28 q^{91} - 10 q^{93} + 84 q^{97} + 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −3.06015 0 −1.22457 0 −1.42687 0 6.36453 0
1.2 0 −2.89159 0 −0.675306 0 −4.40226 0 5.36128 0
1.3 0 −2.76205 0 −3.96400 0 3.92385 0 4.62893 0
1.4 0 −2.22007 0 −0.780901 0 −2.14856 0 1.92871 0
1.5 0 −1.97727 0 −1.88586 0 2.32269 0 0.909580 0
1.6 0 −1.95154 0 3.82321 0 −3.16873 0 0.808502 0
1.7 0 −1.68911 0 1.51721 0 −2.00157 0 −0.146902 0
1.8 0 −1.18907 0 0.375269 0 4.19491 0 −1.58611 0
1.9 0 −1.13155 0 0.498650 0 3.22426 0 −1.71960 0
1.10 0 −0.429134 0 2.14990 0 1.47849 0 −2.81584 0
1.11 0 −0.271416 0 −1.42299 0 2.04647 0 −2.92633 0
1.12 0 −0.0708373 0 3.60899 0 3.08342 0 −2.99498 0
1.13 0 0.0167555 0 3.79279 0 −5.05901 0 −2.99972 0
1.14 0 0.463807 0 −1.33311 0 −3.08911 0 −2.78488 0
1.15 0 0.931158 0 −2.43581 0 −1.58485 0 −2.13295 0
1.16 0 1.21654 0 −2.96314 0 0.724653 0 −1.52003 0
1.17 0 1.52362 0 −2.29226 0 −3.66694 0 −0.678578 0
1.18 0 2.29004 0 2.97542 0 1.34710 0 2.24428 0
1.19 0 2.41359 0 −2.83666 0 3.74478 0 2.82543 0
1.20 0 2.51287 0 1.38190 0 2.25128 0 3.31454 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(29\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6728.2.a.bf 24
29.b even 2 1 6728.2.a.be 24
29.f odd 28 2 232.2.q.a ✓ 48
116.l even 28 2 464.2.y.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.2.q.a ✓ 48 29.f odd 28 2
464.2.y.e 48 116.l even 28 2
6728.2.a.be 24 29.b even 2 1
6728.2.a.bf 24 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6728))\):

\( T_{3}^{24} - 4 T_{3}^{23} - 44 T_{3}^{22} + 178 T_{3}^{21} + 834 T_{3}^{20} - 3382 T_{3}^{19} + \cdots + 64 \) Copy content Toggle raw display
\( T_{5}^{24} - 8 T_{5}^{23} - 44 T_{5}^{22} + 458 T_{5}^{21} + 704 T_{5}^{20} - 11194 T_{5}^{19} + \cdots + 1248073 \) Copy content Toggle raw display