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Properties

Label 1088.2.r.a
Level $1088$
Weight $2$
Character orbit 1088.r
Analytic conductor $8.688$
Analytic rank $0$
Dimension $68$
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] = [1088,2,Mod(305,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1088.305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.r (of order \(4\), 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: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 272)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{13} + 24 q^{15} - 4 q^{17} - 12 q^{19} - 16 q^{21} - 8 q^{33} + 24 q^{35} + 4 q^{43} - 32 q^{47} + 36 q^{49} - 32 q^{51} - 4 q^{53} - 28 q^{59} + 4 q^{67} - 48 q^{69} - 32 q^{77} - 44 q^{81} + 44 q^{83} - 8 q^{85} + 8 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1 0 −2.34859 − 2.34859i 0 −1.90047 + 1.90047i 0 2.22829 0 8.03173i 0
305.2 0 −2.06867 − 2.06867i 0 0.674380 − 0.674380i 0 −1.72950 0 5.55878i 0
305.3 0 −2.06462 − 2.06462i 0 2.55057 − 2.55057i 0 −0.123756 0 5.52535i 0
305.4 0 −1.97734 − 1.97734i 0 −0.636168 + 0.636168i 0 4.57537 0 4.81971i 0
305.5 0 −1.67302 − 1.67302i 0 −2.87951 + 2.87951i 0 −4.68425 0 2.59801i 0
305.6 0 −1.64190 − 1.64190i 0 −0.138160 + 0.138160i 0 −2.35523 0 2.39166i 0
305.7 0 −1.62960 − 1.62960i 0 1.69256 − 1.69256i 0 1.61427 0 2.31119i 0
305.8 0 −1.52926 − 1.52926i 0 −1.18557 + 1.18557i 0 −0.666197 0 1.67726i 0
305.9 0 −1.05504 − 1.05504i 0 1.23969 − 1.23969i 0 3.33560 0 − 0.773762i 0
305.10 0 −0.861322 − 0.861322i 0 0.0304420 − 0.0304420i 0 −2.56029 0 − 1.51625i 0
305.11 0 −0.790194 − 0.790194i 0 −2.80315 + 2.80315i 0 2.35139 0 − 1.75119i 0
305.12 0 −0.783159 − 0.783159i 0 −1.87642 + 1.87642i 0 0.361308 0 − 1.77332i 0
305.13 0 −0.726910 − 0.726910i 0 1.23083 − 1.23083i 0 −0.254563 0 − 1.94320i 0
305.14 0 −0.659351 − 0.659351i 0 1.75731 − 1.75731i 0 4.34538 0 − 2.13051i 0
305.15 0 −0.619466 − 0.619466i 0 −0.496088 + 0.496088i 0 −3.31416 0 − 2.23252i 0
305.16 0 −0.319805 − 0.319805i 0 2.51704 − 2.51704i 0 −1.75502 0 − 2.79545i 0
305.17 0 −0.0355432 − 0.0355432i 0 −1.15813 + 1.15813i 0 3.49333 0 − 2.99747i 0
305.18 0 0.0355432 + 0.0355432i 0 1.15813 − 1.15813i 0 −3.49333 0 − 2.99747i 0
305.19 0 0.319805 + 0.319805i 0 −2.51704 + 2.51704i 0 1.75502 0 − 2.79545i 0
305.20 0 0.619466 + 0.619466i 0 0.496088 − 0.496088i 0 3.31416 0 − 2.23252i 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
17.b even 2 1 inner
272.r even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1088.2.r.a 68
4.b odd 2 1 272.2.r.a ✓ 68
16.e even 4 1 inner 1088.2.r.a 68
16.f odd 4 1 272.2.r.a ✓ 68
17.b even 2 1 inner 1088.2.r.a 68
68.d odd 2 1 272.2.r.a ✓ 68
272.k odd 4 1 272.2.r.a ✓ 68
272.r even 4 1 inner 1088.2.r.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
272.2.r.a ✓ 68 4.b odd 2 1
272.2.r.a ✓ 68 16.f odd 4 1
272.2.r.a ✓ 68 68.d odd 2 1
272.2.r.a ✓ 68 272.k odd 4 1
1088.2.r.a 68 1.a even 1 1 trivial
1088.2.r.a 68 16.e even 4 1 inner
1088.2.r.a 68 17.b even 2 1 inner
1088.2.r.a 68 272.r even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1088, [\chi])\).