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Properties

Label 1288.2.ba.b
Level $1288$
Weight $2$
Character orbit 1288.ba
Analytic conductor $10.285$
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] = [1288,2,Mod(689,1288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1288, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.ba (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: \(10.2847317803\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{3} + 14 q^{9} - 9 q^{23} - 28 q^{25} + 12 q^{29} - 18 q^{31} - 12 q^{35} - 14 q^{39} - 60 q^{47} - 2 q^{49} - 12 q^{59} - 28 q^{71} - 36 q^{73} - 24 q^{75} - 4 q^{77} - 12 q^{81} - 16 q^{85} - 36 q^{87} + 34 q^{93} - 28 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1 0 −2.78229 − 1.60636i 0 −0.838631 − 1.45255i 0 2.08963 + 1.62280i 0 3.66077 + 6.34063i 0
689.2 0 −2.78229 − 1.60636i 0 0.838631 + 1.45255i 0 −2.08963 − 1.62280i 0 3.66077 + 6.34063i 0
689.3 0 −1.62481 − 0.938084i 0 0.737745 + 1.27781i 0 2.63905 − 0.188162i 0 0.260003 + 0.450339i 0
689.4 0 −1.62481 − 0.938084i 0 −0.737745 − 1.27781i 0 −2.63905 + 0.188162i 0 0.260003 + 0.450339i 0
689.5 0 −1.50896 − 0.871200i 0 1.40004 + 2.42493i 0 1.03741 − 2.43388i 0 0.0179791 + 0.0311408i 0
689.6 0 −1.50896 − 0.871200i 0 −1.40004 − 2.42493i 0 −1.03741 + 2.43388i 0 0.0179791 + 0.0311408i 0
689.7 0 −1.21050 − 0.698882i 0 1.80531 + 3.12688i 0 −1.09267 + 2.40958i 0 −0.523129 − 0.906086i 0
689.8 0 −1.21050 − 0.698882i 0 −1.80531 − 3.12688i 0 1.09267 − 2.40958i 0 −0.523129 − 0.906086i 0
689.9 0 −0.620241 − 0.358097i 0 −0.838387 − 1.45213i 0 −1.38077 − 2.25687i 0 −1.24353 − 2.15386i 0
689.10 0 −0.620241 − 0.358097i 0 0.838387 + 1.45213i 0 1.38077 + 2.25687i 0 −1.24353 − 2.15386i 0
689.11 0 0.124737 + 0.0720169i 0 −2.06585 − 3.57815i 0 2.62933 − 0.294356i 0 −1.48963 − 2.58011i 0
689.12 0 0.124737 + 0.0720169i 0 2.06585 + 3.57815i 0 −2.62933 + 0.294356i 0 −1.48963 − 2.58011i 0
689.13 0 0.245596 + 0.141795i 0 0.132905 + 0.230197i 0 −2.25050 − 1.39114i 0 −1.45979 − 2.52843i 0
689.14 0 0.245596 + 0.141795i 0 −0.132905 − 0.230197i 0 2.25050 + 1.39114i 0 −1.45979 − 2.52843i 0
689.15 0 0.890850 + 0.514333i 0 0.297901 + 0.515980i 0 0.458809 − 2.60567i 0 −0.970924 − 1.68169i 0
689.16 0 0.890850 + 0.514333i 0 −0.297901 − 0.515980i 0 −0.458809 + 2.60567i 0 −0.970924 − 1.68169i 0
689.17 0 1.34398 + 0.775948i 0 1.79566 + 3.11018i 0 0.942062 − 2.47235i 0 −0.295810 − 0.512359i 0
689.18 0 1.34398 + 0.775948i 0 −1.79566 − 3.11018i 0 −0.942062 + 2.47235i 0 −0.295810 − 0.512359i 0
689.19 0 1.72539 + 0.996154i 0 0.910097 + 1.57633i 0 1.39371 + 2.24891i 0 0.484645 + 0.839430i 0
689.20 0 1.72539 + 0.996154i 0 −0.910097 − 1.57633i 0 −1.39371 − 2.24891i 0 0.484645 + 0.839430i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 689.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.2.ba.b ✓ 48
7.d odd 6 1 inner 1288.2.ba.b ✓ 48
23.b odd 2 1 inner 1288.2.ba.b ✓ 48
161.g even 6 1 inner 1288.2.ba.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.ba.b ✓ 48 1.a even 1 1 trivial
1288.2.ba.b ✓ 48 7.d odd 6 1 inner
1288.2.ba.b ✓ 48 23.b odd 2 1 inner
1288.2.ba.b ✓ 48 161.g even 6 1 inner