# Properties

 Label 1323.2.a.r 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

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

## 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 −1.00000 0 0 0 0 −2.00000
 $$n$$: 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.r 1
3.b odd 2 1 1323.2.a.b 1
7.b odd 2 1 189.2.a.d yes 1
21.c even 2 1 189.2.a.a 1
28.d even 2 1 3024.2.a.u 1
35.c odd 2 1 4725.2.a.c 1
63.l odd 6 2 567.2.f.a 2
63.o even 6 2 567.2.f.h 2
84.h odd 2 1 3024.2.a.l 1
105.g even 2 1 4725.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.a 1 21.c even 2 1
189.2.a.d yes 1 7.b odd 2 1
567.2.f.a 2 63.l odd 6 2
567.2.f.h 2 63.o even 6 2
1323.2.a.b 1 3.b odd 2 1
1323.2.a.r 1 1.a even 1 1 trivial
3024.2.a.l 1 84.h odd 2 1
3024.2.a.u 1 28.d even 2 1
4725.2.a.c 1 35.c odd 2 1
4725.2.a.s 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

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

## Hecke characteristic polynomials

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