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Properties

Label 1386.2.bu.b
Level $1386$
Weight $2$
Character orbit 1386.bu
Analytic conductor $11.067$
Analytic rank $0$
Dimension $48$
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] = [1386,2,Mod(701,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.701");
 
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.bu (of order \(10\), degree \(4\), 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: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} - 12 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{2} - 12 q^{4} + 12 q^{8} - 4 q^{11} - 12 q^{16} - 24 q^{17} + 4 q^{22} + 24 q^{25} - 40 q^{26} + 16 q^{29} + 40 q^{31} - 48 q^{32} - 16 q^{34} + 12 q^{35} + 16 q^{37} + 40 q^{38} - 24 q^{41} - 4 q^{44} - 40 q^{46} + 40 q^{47} + 12 q^{49} - 4 q^{50} - 40 q^{52} + 40 q^{53} - 32 q^{55} - 16 q^{58} + 40 q^{61} + 40 q^{62} - 12 q^{64} + 48 q^{67} - 24 q^{68} + 8 q^{70} + 40 q^{73} - 16 q^{74} - 32 q^{77} + 40 q^{79} - 16 q^{82} + 16 q^{83} - 20 q^{85} + 4 q^{88} + 20 q^{92} + 52 q^{95} - 8 q^{97} + 48 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1 0.809017 − 0.587785i 0 0.309017 − 0.951057i −2.03332 + 2.79863i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 3.45929i
701.2 0.809017 − 0.587785i 0 0.309017 − 0.951057i −1.25194 + 1.72315i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 2.12993i
701.3 0.809017 − 0.587785i 0 0.309017 − 0.951057i −1.24209 + 1.70960i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 2.11318i
701.4 0.809017 − 0.587785i 0 0.309017 − 0.951057i −1.15641 + 1.59166i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 1.96740i
701.5 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.943712 + 1.29891i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 1.60554i
701.6 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.741879 + 1.02111i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 1.26216i
701.7 0.809017 − 0.587785i 0 0.309017 − 0.951057i −0.0348544 + 0.0479729i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 0.0592978i
701.8 0.809017 − 0.587785i 0 0.309017 − 0.951057i 0.450005 − 0.619379i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 − 0.765594i
701.9 0.809017 − 0.587785i 0 0.309017 − 0.951057i 1.04276 − 1.43524i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 − 1.77406i
701.10 0.809017 − 0.587785i 0 0.309017 − 0.951057i 1.14907 − 1.58155i 0 −0.951057 − 0.309017i −0.309017 − 0.951057i 0 − 1.95491i
701.11 0.809017 − 0.587785i 0 0.309017 − 0.951057i 2.22800 − 3.06659i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 − 3.79051i
701.12 0.809017 − 0.587785i 0 0.309017 − 0.951057i 2.53437 − 3.48826i 0 0.951057 + 0.309017i −0.309017 − 0.951057i 0 − 4.31173i
827.1 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −3.41055 + 1.10815i 0 0.587785 − 0.809017i 0.809017 − 0.587785i 0 − 3.58606i
827.2 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −3.30760 + 1.07471i 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 − 3.47782i
827.3 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −2.76118 + 0.897162i 0 0.587785 − 0.809017i 0.809017 − 0.587785i 0 − 2.90328i
827.4 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −1.68374 + 0.547080i 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 − 1.77039i
827.5 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.678594 + 0.220489i 0 0.587785 − 0.809017i 0.809017 − 0.587785i 0 − 0.713516i
827.6 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.599043 + 0.194641i 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 − 0.629871i
827.7 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i −0.411682 + 0.133764i 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 − 0.432868i
827.8 −0.309017 + 0.951057i 0 −0.809017 − 0.587785i 1.69500 − 0.550738i 0 −0.587785 + 0.809017i 0.809017 − 0.587785i 0 1.78222i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.bu.b yes 48
3.b odd 2 1 1386.2.bu.a ✓ 48
11.d odd 10 1 1386.2.bu.a ✓ 48
33.f even 10 1 inner 1386.2.bu.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.bu.a ✓ 48 3.b odd 2 1
1386.2.bu.a ✓ 48 11.d odd 10 1
1386.2.bu.b yes 48 1.a even 1 1 trivial
1386.2.bu.b yes 48 33.f even 10 1 inner