# Properties

 Label 1323.2.a.bb Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $1$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + 2 \beta_{2} q^{8} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{10} + ( -\beta_{1} - \beta_{2} ) q^{11} + \beta_{1} q^{13} -5 q^{17} + ( -3 \beta_{1} + \beta_{2} ) q^{19} + ( -7 - 3 \beta_{3} ) q^{20} + ( -4 - 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{23} + ( 4 + 4 \beta_{3} ) q^{25} + ( 3 + \beta_{3} ) q^{26} + ( -5 \beta_{1} + 3 \beta_{2} ) q^{29} + ( -\beta_{1} + 4 \beta_{2} ) q^{31} -4 \beta_{2} q^{32} -5 \beta_{1} q^{34} + ( 2 - 3 \beta_{3} ) q^{37} + ( -8 - 2 \beta_{3} ) q^{38} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{40} + ( -2 + 3 \beta_{3} ) q^{41} + \beta_{3} q^{43} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( 3 - \beta_{3} ) q^{46} + ( -3 + 2 \beta_{3} ) q^{47} + ( 8 \beta_{1} + 8 \beta_{2} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{52} + ( 2 \beta_{1} - \beta_{2} ) q^{53} + ( 5 \beta_{1} + 3 \beta_{2} ) q^{55} + ( -12 - 2 \beta_{3} ) q^{58} + ( -5 + 4 \beta_{3} ) q^{59} + ( -\beta_{1} - \beta_{2} ) q^{61} + ( 1 + 3 \beta_{3} ) q^{62} + ( -4 - 4 \beta_{3} ) q^{64} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -9 - \beta_{3} ) q^{67} + ( -5 - 5 \beta_{3} ) q^{68} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -\beta_{1} - 6 \beta_{2} ) q^{74} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{76} + ( -7 - 2 \beta_{3} ) q^{79} + ( \beta_{1} + 6 \beta_{2} ) q^{82} + ( -2 + \beta_{3} ) q^{83} + ( 10 + 5 \beta_{3} ) q^{85} + ( \beta_{1} + 2 \beta_{2} ) q^{86} + ( -6 - 2 \beta_{3} ) q^{88} + ( -5 + 3 \beta_{3} ) q^{89} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -\beta_{1} + 4 \beta_{2} ) q^{94} + ( 7 \beta_{1} + 5 \beta_{2} ) q^{95} + ( \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} - 8q^{5} + O(q^{10})$$ $$4q + 4q^{4} - 8q^{5} - 20q^{17} - 28q^{20} - 16q^{22} + 16q^{25} + 12q^{26} + 8q^{37} - 32q^{38} - 8q^{41} + 12q^{46} - 12q^{47} - 48q^{58} - 20q^{59} + 4q^{62} - 16q^{64} - 36q^{67} - 20q^{68} - 28q^{79} - 8q^{83} + 40q^{85} - 24q^{88} - 20q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 4 \beta_{1}$$

## 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
 −2.28825 −0.874032 0.874032 2.28825
−2.28825 0 3.23607 −4.23607 0 0 −2.82843 0 9.69316
1.2 −0.874032 0 −1.23607 0.236068 0 0 2.82843 0 −0.206331
1.3 0.874032 0 −1.23607 0.236068 0 0 −2.82843 0 0.206331
1.4 2.28825 0 3.23607 −4.23607 0 0 2.82843 0 −9.69316
 $$n$$: 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
21.c even 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.bb 4
3.b odd 2 1 1323.2.a.be yes 4
7.b odd 2 1 1323.2.a.be yes 4
21.c even 2 1 inner 1323.2.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.a.bb 4 1.a even 1 1 trivial
1323.2.a.bb 4 21.c even 2 1 inner
1323.2.a.be yes 4 3.b odd 2 1
1323.2.a.be yes 4 7.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} - 6 T_{2}^{2} + 4$$ $$T_{5}^{2} + 4 T_{5} - 1$$ $$T_{13}^{4} - 6 T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 6 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -1 + 4 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$4 - 14 T^{2} + T^{4}$$
$13$ $$4 - 6 T^{2} + T^{4}$$
$17$ $$( 5 + T )^{4}$$
$19$ $$484 - 46 T^{2} + T^{4}$$
$23$ $$4 - 36 T^{2} + T^{4}$$
$29$ $$3844 - 126 T^{2} + T^{4}$$
$31$ $$484 - 54 T^{2} + T^{4}$$
$37$ $$( -41 - 4 T + T^{2} )^{2}$$
$41$ $$( -41 + 4 T + T^{2} )^{2}$$
$43$ $$( -5 + T^{2} )^{2}$$
$47$ $$( -11 + 6 T + T^{2} )^{2}$$
$53$ $$( -10 + T^{2} )^{2}$$
$59$ $$( -55 + 10 T + T^{2} )^{2}$$
$61$ $$4 - 14 T^{2} + T^{4}$$
$67$ $$( 76 + 18 T + T^{2} )^{2}$$
$71$ $$7744 - 216 T^{2} + T^{4}$$
$73$ $$100 - 180 T^{2} + T^{4}$$
$79$ $$( 29 + 14 T + T^{2} )^{2}$$
$83$ $$( -1 + 4 T + T^{2} )^{2}$$
$89$ $$( -20 + 10 T + T^{2} )^{2}$$
$97$ $$100 - 30 T^{2} + T^{4}$$