# Properties

 Label 1323.2.c.a Level $1323$ Weight $2$ Character orbit 1323.c Analytic conductor $10.564$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} +O(q^{10})$$ $$q + 2 q^{4} + ( 4 - 8 \zeta_{6} ) q^{13} + 4 q^{16} + ( 3 - 6 \zeta_{6} ) q^{19} -5 q^{25} + ( 1 - 2 \zeta_{6} ) q^{31} + 10 q^{37} + 13 q^{43} + ( 8 - 16 \zeta_{6} ) q^{52} + ( -5 + 10 \zeta_{6} ) q^{61} + 8 q^{64} -16 q^{67} + ( -9 + 18 \zeta_{6} ) q^{73} + ( 6 - 12 \zeta_{6} ) q^{76} -4 q^{79} + ( 11 - 22 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{4} + O(q^{10})$$ $$2q + 4q^{4} + 8q^{16} - 10q^{25} + 20q^{37} + 26q^{43} + 16q^{64} - 32q^{67} - 8q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1322.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 2.00000 0 0 0 0 0 0
1322.2 0 0 2.00000 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.c.a 2
3.b odd 2 1 CM 1323.2.c.a 2
7.b odd 2 1 inner 1323.2.c.a 2
7.c even 3 1 189.2.p.a 2
7.d odd 6 1 189.2.p.a 2
21.c even 2 1 inner 1323.2.c.a 2
21.g even 6 1 189.2.p.a 2
21.h odd 6 1 189.2.p.a 2
63.g even 3 1 567.2.s.b 2
63.h even 3 1 567.2.i.a 2
63.i even 6 1 567.2.i.a 2
63.j odd 6 1 567.2.i.a 2
63.k odd 6 1 567.2.s.b 2
63.n odd 6 1 567.2.s.b 2
63.s even 6 1 567.2.s.b 2
63.t odd 6 1 567.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.a 2 7.c even 3 1
189.2.p.a 2 7.d odd 6 1
189.2.p.a 2 21.g even 6 1
189.2.p.a 2 21.h odd 6 1
567.2.i.a 2 63.h even 3 1
567.2.i.a 2 63.i even 6 1
567.2.i.a 2 63.j odd 6 1
567.2.i.a 2 63.t odd 6 1
567.2.s.b 2 63.g even 3 1
567.2.s.b 2 63.k odd 6 1
567.2.s.b 2 63.n odd 6 1
567.2.s.b 2 63.s even 6 1
1323.2.c.a 2 1.a even 1 1 trivial
1323.2.c.a 2 3.b odd 2 1 CM
1323.2.c.a 2 7.b odd 2 1 inner
1323.2.c.a 2 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$27 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( -13 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$75 + T^{2}$$
$67$ $$( 16 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$243 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$363 + T^{2}$$