# Properties

 Label 1512.2.t.d Level $1512$ Weight $2$ Character orbit 1512.t Analytic conductor $12.073$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1512.t (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$22$$ Relative dimension: $$11$$ over $$\Q(\zeta_{3})$$ Twist minimal: no (minimal twist has level 504) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 6q^{5} + 7q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 6q^{5} + 7q^{7} - 6q^{11} - 3q^{13} - 7q^{17} - q^{19} + 4q^{23} + 20q^{25} - 9q^{29} - 4q^{31} - 14q^{35} + 2q^{37} - 16q^{41} - 5q^{47} - 15q^{49} - 11q^{53} + 22q^{55} + 19q^{59} - 13q^{61} - 13q^{65} + 26q^{67} + 48q^{71} - 35q^{73} + 4q^{77} + 10q^{79} + 28q^{83} - 20q^{85} - 6q^{89} - 37q^{91} - 12q^{95} - 29q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1 0 0 0 −3.84095 0 0.676469 + 2.55781i 0 0 0
289.2 0 0 0 −2.66802 0 1.94471 1.79391i 0 0 0
289.3 0 0 0 −1.85591 0 −2.60465 + 0.464545i 0 0 0
289.4 0 0 0 −1.68316 0 0.960133 2.46539i 0 0 0
289.5 0 0 0 −0.340200 0 1.09748 + 2.40739i 0 0 0
289.6 0 0 0 0.481387 0 2.53326 + 0.763277i 0 0 0
289.7 0 0 0 1.58188 0 −1.80922 1.93047i 0 0 0
289.8 0 0 0 1.83657 0 2.45061 0.997255i 0 0 0
289.9 0 0 0 2.52290 0 −1.07705 2.41660i 0 0 0
289.10 0 0 0 3.43592 0 −1.83889 + 1.90223i 0 0 0
289.11 0 0 0 3.52959 0 1.16715 + 2.37440i 0 0 0
361.1 0 0 0 −3.84095 0 0.676469 2.55781i 0 0 0
361.2 0 0 0 −2.66802 0 1.94471 + 1.79391i 0 0 0
361.3 0 0 0 −1.85591 0 −2.60465 0.464545i 0 0 0
361.4 0 0 0 −1.68316 0 0.960133 + 2.46539i 0 0 0
361.5 0 0 0 −0.340200 0 1.09748 2.40739i 0 0 0
361.6 0 0 0 0.481387 0 2.53326 0.763277i 0 0 0
361.7 0 0 0 1.58188 0 −1.80922 + 1.93047i 0 0 0
361.8 0 0 0 1.83657 0 2.45061 + 0.997255i 0 0 0
361.9 0 0 0 2.52290 0 −1.07705 + 2.41660i 0 0 0
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.t.d 22
3.b odd 2 1 504.2.t.d yes 22
4.b odd 2 1 3024.2.t.l 22
7.c even 3 1 1512.2.q.c 22
9.c even 3 1 1512.2.q.c 22
9.d odd 6 1 504.2.q.d 22
12.b even 2 1 1008.2.t.k 22
21.h odd 6 1 504.2.q.d 22
28.g odd 6 1 3024.2.q.k 22
36.f odd 6 1 3024.2.q.k 22
36.h even 6 1 1008.2.q.k 22
63.g even 3 1 inner 1512.2.t.d 22
63.n odd 6 1 504.2.t.d yes 22
84.n even 6 1 1008.2.q.k 22
252.o even 6 1 1008.2.t.k 22
252.bl odd 6 1 3024.2.t.l 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.d 22 9.d odd 6 1
504.2.q.d 22 21.h odd 6 1
504.2.t.d yes 22 3.b odd 2 1
504.2.t.d yes 22 63.n odd 6 1
1008.2.q.k 22 36.h even 6 1
1008.2.q.k 22 84.n even 6 1
1008.2.t.k 22 12.b even 2 1
1008.2.t.k 22 252.o even 6 1
1512.2.q.c 22 7.c even 3 1
1512.2.q.c 22 9.c even 3 1
1512.2.t.d 22 1.a even 1 1 trivial
1512.2.t.d 22 63.g even 3 1 inner
3024.2.q.k 22 28.g odd 6 1
3024.2.q.k 22 36.f odd 6 1
3024.2.t.l 22 4.b odd 2 1
3024.2.t.l 22 252.bl odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{11} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.