# Properties

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

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} -\beta q^{5} -\beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + q^{4} -\beta q^{5} -\beta q^{8} -3 q^{10} -\beta q^{11} -2 q^{13} -5 q^{16} + 4 \beta q^{17} -5 q^{19} -\beta q^{20} -3 q^{22} + \beta q^{23} -2 q^{25} -2 \beta q^{26} -6 \beta q^{29} -5 q^{31} -3 \beta q^{32} + 12 q^{34} -7 q^{37} -5 \beta q^{38} + 3 q^{40} -3 \beta q^{41} -4 q^{43} -\beta q^{44} + 3 q^{46} -4 \beta q^{47} -2 \beta q^{50} -2 q^{52} + 8 \beta q^{53} + 3 q^{55} -18 q^{58} + 4 \beta q^{59} -8 q^{61} -5 \beta q^{62} + q^{64} + 2 \beta q^{65} + 14 q^{67} + 4 \beta q^{68} + 3 \beta q^{71} + 4 q^{73} -7 \beta q^{74} -5 q^{76} + 8 q^{79} + 5 \beta q^{80} -9 q^{82} + 6 \beta q^{83} -12 q^{85} -4 \beta q^{86} + 3 q^{88} + 5 \beta q^{89} + \beta q^{92} -12 q^{94} + 5 \beta q^{95} + 4 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + O(q^{10})$$ $$2q + 2q^{4} - 6q^{10} - 4q^{13} - 10q^{16} - 10q^{19} - 6q^{22} - 4q^{25} - 10q^{31} + 24q^{34} - 14q^{37} + 6q^{40} - 8q^{43} + 6q^{46} - 4q^{52} + 6q^{55} - 36q^{58} - 16q^{61} + 2q^{64} + 28q^{67} + 8q^{73} - 10q^{76} + 16q^{79} - 18q^{82} - 24q^{85} + 6q^{88} - 24q^{94} + 8q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
−1.73205 0 1.00000 1.73205 0 0 1.73205 0 −3.00000
1.2 1.73205 0 1.00000 −1.73205 0 0 −1.73205 0 −3.00000
 $$n$$: 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.t 2
3.b odd 2 1 inner 1323.2.a.t 2
7.b odd 2 1 189.2.a.e 2
21.c even 2 1 189.2.a.e 2
28.d even 2 1 3024.2.a.bg 2
35.c odd 2 1 4725.2.a.ba 2
63.l odd 6 2 567.2.f.k 4
63.o even 6 2 567.2.f.k 4
84.h odd 2 1 3024.2.a.bg 2
105.g even 2 1 4725.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.e 2 7.b odd 2 1
189.2.a.e 2 21.c even 2 1
567.2.f.k 4 63.l odd 6 2
567.2.f.k 4 63.o even 6 2
1323.2.a.t 2 1.a even 1 1 trivial
1323.2.a.t 2 3.b odd 2 1 inner
3024.2.a.bg 2 28.d even 2 1
3024.2.a.bg 2 84.h odd 2 1
4725.2.a.ba 2 35.c odd 2 1
4725.2.a.ba 2 105.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{2} - 3$$ $$T_{5}^{2} - 3$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$-3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-48 + T^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$-3 + T^{2}$$
$29$ $$-108 + T^{2}$$
$31$ $$( 5 + T )^{2}$$
$37$ $$( 7 + T )^{2}$$
$41$ $$-27 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-192 + T^{2}$$
$59$ $$-48 + T^{2}$$
$61$ $$( 8 + T )^{2}$$
$67$ $$( -14 + T )^{2}$$
$71$ $$-27 + T^{2}$$
$73$ $$( -4 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-108 + T^{2}$$
$89$ $$-75 + T^{2}$$
$97$ $$( -4 + T )^{2}$$