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Properties

Label 2688.2.w.d
Level $2688$
Weight $2$
Character orbit 2688.w
Analytic conductor $21.464$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2688,2,Mod(673,2688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2688, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("2688.673");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2688 = 2^{7} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2688.w (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: \(21.4637880633\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 336)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{11} + 8 q^{15} - 8 q^{19} - 4 q^{29} + 24 q^{33} - 4 q^{37} - 20 q^{43} - 28 q^{49} + 8 q^{51} - 20 q^{53} + 40 q^{61} - 28 q^{63} + 16 q^{65} - 4 q^{67} + 16 q^{69} + 16 q^{75} + 4 q^{77} + 24 q^{79} - 28 q^{81} - 40 q^{83} - 48 q^{85} + 72 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1 0 −0.707107 − 0.707107i 0 1.77267 − 1.77267i 0 1.00000i 0 1.00000i 0
673.2 0 −0.707107 − 0.707107i 0 1.84737 − 1.84737i 0 1.00000i 0 1.00000i 0
673.3 0 −0.707107 − 0.707107i 0 0.646579 − 0.646579i 0 1.00000i 0 1.00000i 0
673.4 0 −0.707107 − 0.707107i 0 0.539395 − 0.539395i 0 1.00000i 0 1.00000i 0
673.5 0 −0.707107 − 0.707107i 0 −1.25389 + 1.25389i 0 1.00000i 0 1.00000i 0
673.6 0 −0.707107 − 0.707107i 0 −2.44528 + 2.44528i 0 1.00000i 0 1.00000i 0
673.7 0 −0.707107 − 0.707107i 0 −2.52107 + 2.52107i 0 1.00000i 0 1.00000i 0
673.8 0 0.707107 + 0.707107i 0 3.03597 − 3.03597i 0 1.00000i 0 1.00000i 0
673.9 0 0.707107 + 0.707107i 0 0.805240 − 0.805240i 0 1.00000i 0 1.00000i 0
673.10 0 0.707107 + 0.707107i 0 0.979857 − 0.979857i 0 1.00000i 0 1.00000i 0
673.11 0 0.707107 + 0.707107i 0 −0.116928 + 0.116928i 0 1.00000i 0 1.00000i 0
673.12 0 0.707107 + 0.707107i 0 −2.39875 + 2.39875i 0 1.00000i 0 1.00000i 0
673.13 0 0.707107 + 0.707107i 0 −2.66347 + 2.66347i 0 1.00000i 0 1.00000i 0
673.14 0 0.707107 + 0.707107i 0 1.77230 − 1.77230i 0 1.00000i 0 1.00000i 0
2017.1 0 −0.707107 + 0.707107i 0 1.77267 + 1.77267i 0 − 1.00000i 0 − 1.00000i 0
2017.2 0 −0.707107 + 0.707107i 0 1.84737 + 1.84737i 0 − 1.00000i 0 − 1.00000i 0
2017.3 0 −0.707107 + 0.707107i 0 0.646579 + 0.646579i 0 − 1.00000i 0 − 1.00000i 0
2017.4 0 −0.707107 + 0.707107i 0 0.539395 + 0.539395i 0 − 1.00000i 0 − 1.00000i 0
2017.5 0 −0.707107 + 0.707107i 0 −1.25389 − 1.25389i 0 − 1.00000i 0 − 1.00000i 0
2017.6 0 −0.707107 + 0.707107i 0 −2.44528 − 2.44528i 0 − 1.00000i 0 − 1.00000i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2688.2.w.d 28
4.b odd 2 1 2688.2.w.c 28
8.b even 2 1 336.2.w.b ✓ 28
8.d odd 2 1 1344.2.w.b 28
16.e even 4 1 336.2.w.b ✓ 28
16.e even 4 1 inner 2688.2.w.d 28
16.f odd 4 1 1344.2.w.b 28
16.f odd 4 1 2688.2.w.c 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.w.b ✓ 28 8.b even 2 1
336.2.w.b ✓ 28 16.e even 4 1
1344.2.w.b 28 8.d odd 2 1
1344.2.w.b 28 16.f odd 4 1
2688.2.w.c 28 4.b odd 2 1
2688.2.w.c 28 16.f odd 4 1
2688.2.w.d 28 1.a even 1 1 trivial
2688.2.w.d 28 16.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2688, [\chi])\):

\( T_{5}^{28} - 24 T_{5}^{25} + 560 T_{5}^{24} - 144 T_{5}^{23} + 288 T_{5}^{22} - 13392 T_{5}^{21} + \cdots + 12845056 \) Copy content Toggle raw display
\( T_{11}^{28} - 4 T_{11}^{27} + 8 T_{11}^{26} + 48 T_{11}^{25} + 2364 T_{11}^{24} - 8896 T_{11}^{23} + \cdots + 4091905024 \) Copy content Toggle raw display