# Properties

 Label 1386.2.ba.a Level $1386$ Weight $2$ Character orbit 1386.ba Analytic conductor $11.067$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 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

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{2} - 16q^{4} + 32q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 16q^{2} - 16q^{4} + 32q^{8} - 2q^{11} - 16q^{16} - 4q^{17} + 4q^{22} + 4q^{25} - 16q^{29} + 4q^{31} - 16q^{32} + 8q^{34} - 16q^{35} + 4q^{37} + 32q^{41} - 2q^{44} + 20q^{49} - 8q^{50} - 12q^{55} + 8q^{58} - 8q^{62} + 32q^{64} - 8q^{67} - 4q^{68} - 4q^{70} + 4q^{74} - 14q^{77} - 16q^{82} - 88q^{83} - 2q^{88} + 24q^{95} - 32q^{97} + 8q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
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 1187.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 32
3.b odd 2 1 1386.2.ba.b yes 32
7.c even 3 1 inner 1386.2.ba.a 32
11.b odd 2 1 1386.2.ba.b yes 32
21.h odd 6 1 1386.2.ba.b yes 32
33.d even 2 1 inner 1386.2.ba.a 32
77.h odd 6 1 1386.2.ba.b yes 32
231.l even 6 1 inner 1386.2.ba.a 32

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