# Properties

 Label 1323.2.a.s Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 2q^{4} + 3q^{5} + O(q^{10})$$ $$q + 2q^{2} + 2q^{4} + 3q^{5} + 6q^{10} + 2q^{11} + 6q^{13} - 4q^{16} + 3q^{17} - 6q^{19} + 6q^{20} + 4q^{22} - 8q^{23} + 4q^{25} + 12q^{26} - 2q^{29} + 6q^{31} - 8q^{32} + 6q^{34} + 9q^{37} - 12q^{38} + 9q^{41} - 9q^{43} + 4q^{44} - 16q^{46} - 3q^{47} + 8q^{50} + 12q^{52} + 4q^{53} + 6q^{55} - 4q^{58} - 3q^{59} - 6q^{61} + 12q^{62} - 8q^{64} + 18q^{65} + 4q^{67} + 6q^{68} - 4q^{71} - 12q^{73} + 18q^{74} - 12q^{76} - q^{79} - 12q^{80} + 18q^{82} + 15q^{83} + 9q^{85} - 18q^{86} - 6q^{89} - 16q^{92} - 6q^{94} - 18q^{95} - 6q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
2.00000 0 2.00000 3.00000 0 0 0 0 6.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.s yes 1
3.b odd 2 1 1323.2.a.a 1
7.b odd 2 1 1323.2.a.q yes 1
21.c even 2 1 1323.2.a.c yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.a.a 1 3.b odd 2 1
1323.2.a.c yes 1 21.c even 2 1
1323.2.a.q yes 1 7.b odd 2 1
1323.2.a.s yes 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2} - 2$$ $$T_{5} - 3$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$-2 + T$$
$13$ $$-6 + T$$
$17$ $$-3 + T$$
$19$ $$6 + T$$
$23$ $$8 + T$$
$29$ $$2 + T$$
$31$ $$-6 + T$$
$37$ $$-9 + T$$
$41$ $$-9 + T$$
$43$ $$9 + T$$
$47$ $$3 + T$$
$53$ $$-4 + T$$
$59$ $$3 + T$$
$61$ $$6 + T$$
$67$ $$-4 + T$$
$71$ $$4 + T$$
$73$ $$12 + T$$
$79$ $$1 + T$$
$83$ $$-15 + T$$
$89$ $$6 + T$$
$97$ $$6 + T$$