# Properties

 Label 1512.2.t.c 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.

 $$\operatorname{Tr}(f)(q) =$$ $$22q + 2q^{5} - q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$22q + 2q^{5} - q^{7} + 6q^{11} + 7q^{13} + q^{17} + 13q^{19} + 44q^{25} + 7q^{29} + 6q^{31} - 2q^{35} + 6q^{37} - 4q^{41} + 2q^{43} - 17q^{47} + 29q^{49} - q^{53} + 2q^{55} + 21q^{59} + 31q^{61} + 3q^{65} - 26q^{67} + 32q^{71} + 17q^{73} + 4q^{77} - 16q^{79} + 36q^{83} + 28q^{85} + 2q^{89} + 15q^{91} + 24q^{95} + 19q^{97} + O(q^{100})$$

## 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 −3.40736 0 −2.05842 + 1.66220i 0 0 0
289.2 0 0 0 −3.19500 0 2.61289 + 0.415693i 0 0 0
289.3 0 0 0 −2.77180 0 0.855737 2.50354i 0 0 0
289.4 0 0 0 −2.10440 0 −1.78475 1.95312i 0 0 0
289.5 0 0 0 −0.526004 0 2.43963 1.02383i 0 0 0
289.6 0 0 0 0.0619693 0 −1.63689 + 2.07860i 0 0 0
289.7 0 0 0 0.468169 0 −2.39007 + 1.13471i 0 0 0
289.8 0 0 0 1.78355 0 1.90167 + 1.83948i 0 0 0
289.9 0 0 0 2.66851 0 −0.654882 2.56342i 0 0 0
289.10 0 0 0 3.79940 0 2.59312 0.525101i 0 0 0
289.11 0 0 0 4.22296 0 −2.37802 1.15974i 0 0 0
361.1 0 0 0 −3.40736 0 −2.05842 1.66220i 0 0 0
361.2 0 0 0 −3.19500 0 2.61289 0.415693i 0 0 0
361.3 0 0 0 −2.77180 0 0.855737 + 2.50354i 0 0 0
361.4 0 0 0 −2.10440 0 −1.78475 + 1.95312i 0 0 0
361.5 0 0 0 −0.526004 0 2.43963 + 1.02383i 0 0 0
361.6 0 0 0 0.0619693 0 −1.63689 2.07860i 0 0 0
361.7 0 0 0 0.468169 0 −2.39007 1.13471i 0 0 0
361.8 0 0 0 1.78355 0 1.90167 1.83948i 0 0 0
361.9 0 0 0 2.66851 0 −0.654882 + 2.56342i 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.c 22
3.b odd 2 1 504.2.t.c yes 22
4.b odd 2 1 3024.2.t.k 22
7.c even 3 1 1512.2.q.d 22
9.c even 3 1 1512.2.q.d 22
9.d odd 6 1 504.2.q.c 22
12.b even 2 1 1008.2.t.l 22
21.h odd 6 1 504.2.q.c 22
28.g odd 6 1 3024.2.q.l 22
36.f odd 6 1 3024.2.q.l 22
36.h even 6 1 1008.2.q.l 22
63.g even 3 1 inner 1512.2.t.c 22
63.n odd 6 1 504.2.t.c yes 22
84.n even 6 1 1008.2.q.l 22
252.o even 6 1 1008.2.t.l 22
252.bl odd 6 1 3024.2.t.k 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.c 22 9.d odd 6 1
504.2.q.c 22 21.h odd 6 1
504.2.t.c yes 22 3.b odd 2 1
504.2.t.c yes 22 63.n odd 6 1
1008.2.q.l 22 36.h even 6 1
1008.2.q.l 22 84.n even 6 1
1008.2.t.l 22 12.b even 2 1
1008.2.t.l 22 252.o even 6 1
1512.2.q.d 22 7.c even 3 1
1512.2.q.d 22 9.c even 3 1
1512.2.t.c 22 1.a even 1 1 trivial
1512.2.t.c 22 63.g even 3 1 inner
3024.2.q.l 22 28.g odd 6 1
3024.2.q.l 22 36.f odd 6 1
3024.2.t.k 22 4.b odd 2 1
3024.2.t.k 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 $$S_{2}^{\mathrm{new}}(1512, [\chi])$$.