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Properties

Label 1288.2.ba.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{3} + 30 q^{9} - q^{23} - 28 q^{25} + 12 q^{29} + 18 q^{31} + 32 q^{35} - 6 q^{39} + 24 q^{47} + 54 q^{49} - 12 q^{59} + 28 q^{71} + 12 q^{73} + 72 q^{75} - 8 q^{77} - 44 q^{81} - 40 q^{85} - 72 q^{87} + 38 q^{93} + 12 q^{95}+O(q^{100}) \) 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 −2.78163 − 1.60598i 0 2.09422 + 3.62729i 0 2.64268 + 0.127468i 0 3.65832 + 6.33640i 0
689.2 0 −2.78163 − 1.60598i 0 −2.09422 − 3.62729i 0 −2.64268 − 0.127468i 0 3.65832 + 6.33640i 0
689.3 0 −2.33262 − 1.34674i 0 0.733022 + 1.26963i 0 −0.944201 + 2.47153i 0 2.12741 + 3.68479i 0
689.4 0 −2.33262 − 1.34674i 0 −0.733022 − 1.26963i 0 0.944201 − 2.47153i 0 2.12741 + 3.68479i 0
689.5 0 −2.06222 − 1.19062i 0 −0.669431 − 1.15949i 0 1.00087 + 2.44913i 0 1.33516 + 2.31256i 0
689.6 0 −2.06222 − 1.19062i 0 0.669431 + 1.15949i 0 −1.00087 − 2.44913i 0 1.33516 + 2.31256i 0
689.7 0 −1.34405 − 0.775987i 0 1.27086 + 2.20119i 0 −2.48009 − 0.921490i 0 −0.295689 − 0.512148i 0
689.8 0 −1.34405 − 0.775987i 0 −1.27086 − 2.20119i 0 2.48009 + 0.921490i 0 −0.295689 − 0.512148i 0
689.9 0 −1.10379 − 0.637271i 0 0.555777 + 0.962634i 0 −2.63443 + 0.244514i 0 −0.687771 − 1.19125i 0
689.10 0 −1.10379 − 0.637271i 0 −0.555777 − 0.962634i 0 2.63443 − 0.244514i 0 −0.687771 − 1.19125i 0
689.11 0 −0.358903 − 0.207213i 0 −1.59352 − 2.76006i 0 −2.64573 + 0.0114901i 0 −1.41413 − 2.44934i 0
689.12 0 −0.358903 − 0.207213i 0 1.59352 + 2.76006i 0 2.64573 − 0.0114901i 0 −1.41413 − 2.44934i 0
689.13 0 −0.114538 − 0.0661285i 0 −1.46504 − 2.53753i 0 −0.141894 − 2.64194i 0 −1.49125 − 2.58293i 0
689.14 0 −0.114538 − 0.0661285i 0 1.46504 + 2.53753i 0 0.141894 + 2.64194i 0 −1.49125 − 2.58293i 0
689.15 0 0.511589 + 0.295366i 0 0.229601 + 0.397681i 0 1.39992 − 2.24504i 0 −1.32552 − 2.29586i 0
689.16 0 0.511589 + 0.295366i 0 −0.229601 − 0.397681i 0 −1.39992 + 2.24504i 0 −1.32552 − 2.29586i 0
689.17 0 1.62739 + 0.939573i 0 1.57271 + 2.72401i 0 2.50259 − 0.858509i 0 0.265597 + 0.460027i 0
689.18 0 1.62739 + 0.939573i 0 −1.57271 − 2.72401i 0 −2.50259 + 0.858509i 0 0.265597 + 0.460027i 0
689.19 0 1.67410 + 0.966544i 0 0.0661174 + 0.114519i 0 −1.78409 − 1.95372i 0 0.368413 + 0.638111i 0
689.20 0 1.67410 + 0.966544i 0 −0.0661174 − 0.114519i 0 1.78409 + 1.95372i 0 0.368413 + 0.638111i 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.a ✓ 48
7.d odd 6 1 inner 1288.2.ba.a ✓ 48
23.b odd 2 1 inner 1288.2.ba.a ✓ 48
161.g even 6 1 inner 1288.2.ba.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.ba.a ✓ 48 1.a even 1 1 trivial
1288.2.ba.a ✓ 48 7.d odd 6 1 inner
1288.2.ba.a ✓ 48 23.b odd 2 1 inner
1288.2.ba.a ✓ 48 161.g even 6 1 inner