# 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 $2$

# Related objects

## Newspace parameters

 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

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 12q^{2} - 12q^{4} + 12q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 12q^{2} - 12q^{4} + 12q^{8} - 4q^{11} - 12q^{16} - 24q^{17} + 4q^{22} + 24q^{25} - 40q^{26} + 16q^{29} + 40q^{31} - 48q^{32} - 16q^{34} + 12q^{35} + 16q^{37} + 40q^{38} - 24q^{41} - 4q^{44} - 40q^{46} + 40q^{47} + 12q^{49} - 4q^{50} - 40q^{52} + 40q^{53} - 32q^{55} - 16q^{58} + 40q^{61} + 40q^{62} - 12q^{64} + 48q^{67} - 24q^{68} + 8q^{70} + 40q^{73} - 16q^{74} - 32q^{77} + 40q^{79} - 16q^{82} + 16q^{83} - 20q^{85} + 4q^{88} + 20q^{92} + 52q^{95} - 8q^{97} + 48q^{98} + O(q^{100})$$

## 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.

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 1205.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database