# Properties

 Label 1386.2.bk.d Level $1386$ Weight $2$ Character orbit 1386.bk Analytic conductor $11.067$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.bk (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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{4} - 16q^{16} + 20q^{22} + 36q^{25} + 12q^{31} - 20q^{37} - 44q^{49} - 32q^{64} + 48q^{67} + 36q^{70} + 72q^{82} + 10q^{88} + 144q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.194145 + 0.112089i 0 2.62785 + 0.307218i 1.00000i 0 0.112089 0.194145i
703.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.194145 0.112089i 0 −2.62785 0.307218i 1.00000i 0 −0.112089 + 0.194145i
703.3 −0.866025 + 0.500000i 0 0.500000 0.866025i −1.23351 + 0.712169i 0 −0.484566 2.60100i 1.00000i 0 0.712169 1.23351i
703.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.23351 0.712169i 0 0.484566 + 2.60100i 1.00000i 0 −0.712169 + 1.23351i
703.5 −0.866025 + 0.500000i 0 0.500000 0.866025i −3.71337 + 2.14392i 0 −0.293638 + 2.62941i 1.00000i 0 2.14392 3.71337i
703.6 −0.866025 + 0.500000i 0 0.500000 0.866025i 3.71337 2.14392i 0 0.293638 2.62941i 1.00000i 0 −2.14392 + 3.71337i
703.7 −0.866025 + 0.500000i 0 0.500000 0.866025i −2.53014 + 1.46078i 0 2.00583 1.72530i 1.00000i 0 1.46078 2.53014i
703.8 −0.866025 + 0.500000i 0 0.500000 0.866025i 2.53014 1.46078i 0 −2.00583 + 1.72530i 1.00000i 0 −1.46078 + 2.53014i
703.9 0.866025 0.500000i 0 0.500000 0.866025i −0.194145 + 0.112089i 0 −2.62785 0.307218i 1.00000i 0 −0.112089 + 0.194145i
703.10 0.866025 0.500000i 0 0.500000 0.866025i 0.194145 0.112089i 0 2.62785 + 0.307218i 1.00000i 0 0.112089 0.194145i
703.11 0.866025 0.500000i 0 0.500000 0.866025i −2.53014 + 1.46078i 0 −2.00583 + 1.72530i 1.00000i 0 −1.46078 + 2.53014i
703.12 0.866025 0.500000i 0 0.500000 0.866025i 2.53014 1.46078i 0 2.00583 1.72530i 1.00000i 0 1.46078 2.53014i
703.13 0.866025 0.500000i 0 0.500000 0.866025i −1.23351 + 0.712169i 0 0.484566 + 2.60100i 1.00000i 0 −0.712169 + 1.23351i
703.14 0.866025 0.500000i 0 0.500000 0.866025i 1.23351 0.712169i 0 −0.484566 2.60100i 1.00000i 0 0.712169 1.23351i
703.15 0.866025 0.500000i 0 0.500000 0.866025i −3.71337 + 2.14392i 0 0.293638 2.62941i 1.00000i 0 −2.14392 + 3.71337i
703.16 0.866025 0.500000i 0 0.500000 0.866025i 3.71337 2.14392i 0 −0.293638 + 2.62941i 1.00000i 0 2.14392 3.71337i
901.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.194145 0.112089i 0 2.62785 0.307218i 1.00000i 0 0.112089 + 0.194145i
901.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.194145 + 0.112089i 0 −2.62785 + 0.307218i 1.00000i 0 −0.112089 0.194145i
901.3 −0.866025 0.500000i 0 0.500000 + 0.866025i −1.23351 0.712169i 0 −0.484566 + 2.60100i 1.00000i 0 0.712169 + 1.23351i
901.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.23351 + 0.712169i 0 0.484566 2.60100i 1.00000i 0 −0.712169 1.23351i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
21.g even 6 1 inner
33.d even 2 1 inner
77.i even 6 1 inner
231.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.bk.d 32
3.b odd 2 1 inner 1386.2.bk.d 32
7.d odd 6 1 inner 1386.2.bk.d 32
11.b odd 2 1 inner 1386.2.bk.d 32
21.g even 6 1 inner 1386.2.bk.d 32
33.d even 2 1 inner 1386.2.bk.d 32
77.i even 6 1 inner 1386.2.bk.d 32
231.k odd 6 1 inner 1386.2.bk.d 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.bk.d 32 1.a even 1 1 trivial
1386.2.bk.d 32 3.b odd 2 1 inner
1386.2.bk.d 32 7.d odd 6 1 inner
1386.2.bk.d 32 11.b odd 2 1 inner
1386.2.bk.d 32 21.g even 6 1 inner
1386.2.bk.d 32 33.d even 2 1 inner
1386.2.bk.d 32 77.i even 6 1 inner
1386.2.bk.d 32 231.k odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$:

 $$T_{5}^{16} - \cdots$$ $$T_{13}^{8} - 56 T_{13}^{6} + 948 T_{13}^{4} - 4496 T_{13}^{2} + 4096$$

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database