# Properties

 Label 1323.2.a.p 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: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} + 4q^{5} - 3q^{8} + O(q^{10})$$ $$q + q^{2} - q^{4} + 4q^{5} - 3q^{8} + 4q^{10} + 2q^{11} - q^{13} - q^{16} + 6q^{17} - 4q^{19} - 4q^{20} + 2q^{22} + 6q^{23} + 11q^{25} - q^{26} + 2q^{29} - 3q^{31} + 5q^{32} + 6q^{34} + 3q^{37} - 4q^{38} - 12q^{40} + 2q^{41} - q^{43} - 2q^{44} + 6q^{46} - 6q^{47} + 11q^{50} + q^{52} - 6q^{53} + 8q^{55} + 2q^{58} - 6q^{59} + 5q^{61} - 3q^{62} + 7q^{64} - 4q^{65} + 7q^{67} - 6q^{68} + 6q^{73} + 3q^{74} + 4q^{76} + 11q^{79} - 4q^{80} + 2q^{82} - 6q^{83} + 24q^{85} - q^{86} - 6q^{88} + 4q^{89} - 6q^{92} - 6q^{94} - 16q^{95} - 9q^{97} + O(q^{100})$$

## 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
1.00000 0 −1.00000 4.00000 0 0 −3.00000 0 4.00000
 $$n$$: 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.p 1
3.b odd 2 1 1323.2.a.d 1
7.b odd 2 1 1323.2.a.m 1
7.d odd 6 2 189.2.e.a 2
21.c even 2 1 1323.2.a.g 1
21.g even 6 2 189.2.e.c yes 2
63.i even 6 2 567.2.h.b 2
63.k odd 6 2 567.2.g.b 2
63.s even 6 2 567.2.g.e 2
63.t odd 6 2 567.2.h.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.a 2 7.d odd 6 2
189.2.e.c yes 2 21.g even 6 2
567.2.g.b 2 63.k odd 6 2
567.2.g.e 2 63.s even 6 2
567.2.h.b 2 63.i even 6 2
567.2.h.e 2 63.t odd 6 2
1323.2.a.d 1 3.b odd 2 1
1323.2.a.g 1 21.c even 2 1
1323.2.a.m 1 7.b odd 2 1
1323.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2} - 1$$ $$T_{5} - 4$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

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