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Properties

Label 1386.2.ba.b
Level $1386$
Weight $2$
Character orbit 1386.ba
Analytic conductor $11.067$
Analytic rank $0$
Dimension $32$
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] = [1386,2,Mod(989,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.989");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{2} - 16 q^{4} - 32 q^{8} + 2 q^{11} - 16 q^{16} + 4 q^{17} + 4 q^{22} + 4 q^{25} + 16 q^{29} + 4 q^{31} + 16 q^{32} + 8 q^{34} + 16 q^{35} + 4 q^{37} - 32 q^{41} + 2 q^{44} + 20 q^{49} + 8 q^{50} - 12 q^{55} + 8 q^{58} + 8 q^{62} + 32 q^{64} - 8 q^{67} + 4 q^{68} - 4 q^{70} - 4 q^{74} + 14 q^{77} - 16 q^{82} + 88 q^{83} - 2 q^{88} - 24 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
989.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −3.19684 − 1.84570i 0 1.52985 + 2.15860i −1.00000 0 −3.19684 + 1.84570i
989.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.42143 − 1.39801i 0 −2.62875 − 0.299420i −1.00000 0 −2.42143 + 1.39801i
989.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.34835 − 1.35582i 0 −0.222226 − 2.63640i −1.00000 0 −2.34835 + 1.35582i
989.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.87061 − 1.08000i 0 −2.63666 + 0.219149i −1.00000 0 −1.87061 + 1.08000i
989.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.78334 − 1.02961i 0 −0.289722 + 2.62984i −1.00000 0 −1.78334 + 1.02961i
989.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.42385 − 0.822059i 0 2.58759 − 0.551706i −1.00000 0 −1.42385 + 0.822059i
989.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.03699 − 0.598709i 0 2.52745 − 0.782297i −1.00000 0 −1.03699 + 0.598709i
989.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.346303 − 0.199938i 0 −1.03937 + 2.43304i −1.00000 0 −0.346303 + 0.199938i
989.9 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.346303 + 0.199938i 0 1.03937 − 2.43304i −1.00000 0 0.346303 − 0.199938i
989.10 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.03699 + 0.598709i 0 −2.52745 + 0.782297i −1.00000 0 1.03699 − 0.598709i
989.11 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.42385 + 0.822059i 0 −2.58759 + 0.551706i −1.00000 0 1.42385 − 0.822059i
989.12 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.78334 + 1.02961i 0 0.289722 − 2.62984i −1.00000 0 1.78334 − 1.02961i
989.13 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.87061 + 1.08000i 0 2.63666 − 0.219149i −1.00000 0 1.87061 − 1.08000i
989.14 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.34835 + 1.35582i 0 0.222226 + 2.63640i −1.00000 0 2.34835 − 1.35582i
989.15 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.42143 + 1.39801i 0 2.62875 + 0.299420i −1.00000 0 2.42143 − 1.39801i
989.16 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 3.19684 + 1.84570i 0 −1.52985 − 2.15860i −1.00000 0 3.19684 − 1.84570i
1187.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.19684 + 1.84570i 0 1.52985 − 2.15860i −1.00000 0 −3.19684 − 1.84570i
1187.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.42143 + 1.39801i 0 −2.62875 + 0.299420i −1.00000 0 −2.42143 − 1.39801i
1187.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.34835 + 1.35582i 0 −0.222226 + 2.63640i −1.00000 0 −2.34835 − 1.35582i
1187.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.87061 + 1.08000i 0 −2.63666 − 0.219149i −1.00000 0 −1.87061 − 1.08000i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
33.d even 2 1 inner
231.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.ba.b yes 32
3.b odd 2 1 1386.2.ba.a ✓ 32
7.c even 3 1 inner 1386.2.ba.b yes 32
11.b odd 2 1 1386.2.ba.a ✓ 32
21.h odd 6 1 1386.2.ba.a ✓ 32
33.d even 2 1 inner 1386.2.ba.b yes 32
77.h odd 6 1 1386.2.ba.a ✓ 32
231.l even 6 1 inner 1386.2.ba.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.ba.a ✓ 32 3.b odd 2 1
1386.2.ba.a ✓ 32 11.b odd 2 1
1386.2.ba.a ✓ 32 21.h odd 6 1
1386.2.ba.a ✓ 32 77.h odd 6 1
1386.2.ba.b yes 32 1.a even 1 1 trivial
1386.2.ba.b yes 32 7.c even 3 1 inner
1386.2.ba.b yes 32 33.d even 2 1 inner
1386.2.ba.b yes 32 231.l even 6 1 inner