# Properties

 Label 1323.2.a.l 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

 $$f(q)$$ $$=$$ $$q - 2q^{4} + 3q^{5} + O(q^{10})$$ $$q - 2q^{4} + 3q^{5} - 6q^{11} + 4q^{13} + 4q^{16} + 3q^{17} - 2q^{19} - 6q^{20} + 6q^{23} + 4q^{25} + 6q^{29} + 4q^{31} - 7q^{37} - 3q^{41} - q^{43} + 12q^{44} + 9q^{47} - 8q^{52} + 6q^{53} - 18q^{55} + 9q^{59} + 10q^{61} - 8q^{64} + 12q^{65} - 4q^{67} - 6q^{68} - 2q^{73} + 4q^{76} - q^{79} + 12q^{80} + 3q^{83} + 9q^{85} + 6q^{89} - 12q^{92} - 6q^{95} + 10q^{97} + O(q^{100})$$

## 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
0 0 −2.00000 3.00000 0 0 0 0 0
 $$n$$: 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.l 1
3.b odd 2 1 1323.2.a.h 1
7.b odd 2 1 189.2.a.b 1
21.c even 2 1 189.2.a.c yes 1
28.d even 2 1 3024.2.a.f 1
35.c odd 2 1 4725.2.a.i 1
63.l odd 6 2 567.2.f.e 2
63.o even 6 2 567.2.f.d 2
84.h odd 2 1 3024.2.a.y 1
105.g even 2 1 4725.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.b 1 7.b odd 2 1
189.2.a.c yes 1 21.c even 2 1
567.2.f.d 2 63.o even 6 2
567.2.f.e 2 63.l odd 6 2
1323.2.a.h 1 3.b odd 2 1
1323.2.a.l 1 1.a even 1 1 trivial
3024.2.a.f 1 28.d even 2 1
3024.2.a.y 1 84.h odd 2 1
4725.2.a.i 1 35.c odd 2 1
4725.2.a.k 1 105.g even 2 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{5} - 3$$ $$T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-3 + T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$-4 + T$$
$17$ $$-3 + T$$
$19$ $$2 + T$$
$23$ $$-6 + T$$
$29$ $$-6 + T$$
$31$ $$-4 + T$$
$37$ $$7 + T$$
$41$ $$3 + T$$
$43$ $$1 + T$$
$47$ $$-9 + T$$
$53$ $$-6 + T$$
$59$ $$-9 + T$$
$61$ $$-10 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$1 + T$$
$83$ $$-3 + T$$
$89$ $$-6 + T$$
$97$ $$-10 + T$$