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Properties

Label 1288.2.q.a
Level $1288$
Weight $2$
Character orbit 1288.q
Analytic conductor $10.285$
Analytic rank $0$
Dimension $22$
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] = [1288,2,Mod(737,1288)]
 
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, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1288.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.q (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{3} - q^{5} + q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 4 q^{3} - q^{5} + q^{7} - 11 q^{9} - 6 q^{13} - 16 q^{15} - 5 q^{17} - 12 q^{19} + 3 q^{21} - 11 q^{23} - 22 q^{25} + 38 q^{27} - 30 q^{29} - 16 q^{31} - 4 q^{33} - 5 q^{35} - 3 q^{37} + q^{39} + 56 q^{41} - 18 q^{43} + 19 q^{45} - 31 q^{47} + 13 q^{49} + 15 q^{51} - 13 q^{53} + 70 q^{55} - 42 q^{57} - 11 q^{59} + 19 q^{61} - 23 q^{63} + 7 q^{65} - 19 q^{67} + 8 q^{69} - 10 q^{71} + 5 q^{73} - 28 q^{75} + 9 q^{77} + 13 q^{79} - 35 q^{81} + 34 q^{83} - 78 q^{85} - 4 q^{87} - 10 q^{89} - 30 q^{91} + 6 q^{93} - 33 q^{95} + 70 q^{97} + 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1 0 −1.56813 − 2.71608i 0 0.415569 − 0.719787i 0 −1.83313 + 1.90779i 0 −3.41805 + 5.92023i 0
737.2 0 −1.46797 − 2.54261i 0 −1.03420 + 1.79128i 0 2.64527 + 0.0504627i 0 −2.80990 + 4.86688i 0
737.3 0 −1.42915 − 2.47536i 0 1.92193 − 3.32888i 0 1.69651 − 2.03023i 0 −2.58493 + 4.47724i 0
737.4 0 −0.473547 − 0.820207i 0 −1.73047 + 2.99726i 0 −2.32170 + 1.26874i 0 1.05151 − 1.82126i 0
737.5 0 −0.328152 − 0.568377i 0 1.04571 − 1.81122i 0 −2.32073 − 1.27052i 0 1.28463 − 2.22505i 0
737.6 0 −0.244193 − 0.422955i 0 1.32227 − 2.29024i 0 −1.16133 − 2.37725i 0 1.38074 − 2.39151i 0
737.7 0 −0.0444961 − 0.0770695i 0 −0.912579 + 1.58063i 0 1.82687 + 1.91377i 0 1.49604 − 2.59122i 0
737.8 0 0.203096 + 0.351772i 0 −2.21789 + 3.84149i 0 0.196067 − 2.63848i 0 1.41750 − 2.45519i 0
737.9 0 0.798877 + 1.38370i 0 1.53553 − 2.65962i 0 1.24883 + 2.33247i 0 0.223590 − 0.387270i 0
737.10 0 1.20625 + 2.08929i 0 −0.352012 + 0.609702i 0 2.64041 − 0.168024i 0 −1.41008 + 2.44234i 0
737.11 0 1.34741 + 2.33379i 0 −0.493872 + 0.855412i 0 −2.11708 − 1.58681i 0 −2.13105 + 3.69109i 0
921.1 0 −1.56813 + 2.71608i 0 0.415569 + 0.719787i 0 −1.83313 − 1.90779i 0 −3.41805 − 5.92023i 0
921.2 0 −1.46797 + 2.54261i 0 −1.03420 − 1.79128i 0 2.64527 − 0.0504627i 0 −2.80990 − 4.86688i 0
921.3 0 −1.42915 + 2.47536i 0 1.92193 + 3.32888i 0 1.69651 + 2.03023i 0 −2.58493 − 4.47724i 0
921.4 0 −0.473547 + 0.820207i 0 −1.73047 − 2.99726i 0 −2.32170 − 1.26874i 0 1.05151 + 1.82126i 0
921.5 0 −0.328152 + 0.568377i 0 1.04571 + 1.81122i 0 −2.32073 + 1.27052i 0 1.28463 + 2.22505i 0
921.6 0 −0.244193 + 0.422955i 0 1.32227 + 2.29024i 0 −1.16133 + 2.37725i 0 1.38074 + 2.39151i 0
921.7 0 −0.0444961 + 0.0770695i 0 −0.912579 − 1.58063i 0 1.82687 − 1.91377i 0 1.49604 + 2.59122i 0
921.8 0 0.203096 − 0.351772i 0 −2.21789 − 3.84149i 0 0.196067 + 2.63848i 0 1.41750 + 2.45519i 0
921.9 0 0.798877 − 1.38370i 0 1.53553 + 2.65962i 0 1.24883 − 2.33247i 0 0.223590 + 0.387270i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1288.2.q.a ✓ 22
7.c even 3 1 inner 1288.2.q.a ✓ 22
7.c even 3 1 9016.2.a.bq 11
7.d odd 6 1 9016.2.a.bl 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1288.2.q.a ✓ 22 1.a even 1 1 trivial
1288.2.q.a ✓ 22 7.c even 3 1 inner
9016.2.a.bl 11 7.d odd 6 1
9016.2.a.bq 11 7.c even 3 1