# Properties

 Label 1323.2.a.bd Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7168.1 Defining polynomial: $$x^{4} - 6 x^{2} + 7$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{1} q^{5} + \beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{1} q^{5} + \beta_{3} q^{8} + ( 3 + \beta_{2} ) q^{10} + ( -\beta_{1} - 2 \beta_{3} ) q^{11} + ( 4 + 2 \beta_{2} ) q^{13} -3 q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + ( 2 \beta_{1} + \beta_{3} ) q^{20} + ( -1 - 5 \beta_{2} ) q^{22} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{23} + ( -2 + \beta_{2} ) q^{25} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{26} + 4 \beta_{1} q^{29} + ( 5 - 4 \beta_{2} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( 4 - \beta_{2} ) q^{34} + ( -1 + 3 \beta_{2} ) q^{37} + ( 4 \beta_{1} + \beta_{3} ) q^{38} + ( -1 + 2 \beta_{2} ) q^{40} + ( -2 \beta_{1} - \beta_{3} ) q^{41} -5 \beta_{2} q^{43} + ( -4 \beta_{1} - \beta_{3} ) q^{44} + ( 3 + 8 \beta_{2} ) q^{46} + ( -3 \beta_{1} - 5 \beta_{3} ) q^{47} + ( -\beta_{1} + \beta_{3} ) q^{50} + ( 8 + 6 \beta_{2} ) q^{52} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{53} + ( -1 - 5 \beta_{2} ) q^{55} + ( 12 + 4 \beta_{2} ) q^{58} + ( \beta_{1} + \beta_{3} ) q^{59} + ( 2 - 6 \beta_{2} ) q^{61} + ( \beta_{1} - 4 \beta_{3} ) q^{62} + ( -1 - 7 \beta_{2} ) q^{64} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{65} -4 \beta_{2} q^{67} + ( \beta_{1} + \beta_{3} ) q^{68} -7 \beta_{1} q^{71} + ( 10 + \beta_{2} ) q^{73} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{74} + ( 5 + 4 \beta_{2} ) q^{76} + ( -4 + 7 \beta_{2} ) q^{79} -3 \beta_{1} q^{80} + ( -5 - 4 \beta_{2} ) q^{82} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{83} + ( 4 - \beta_{2} ) q^{85} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{86} + ( -9 + 4 \beta_{2} ) q^{88} + ( -2 \beta_{1} + 5 \beta_{3} ) q^{89} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -4 - 13 \beta_{2} ) q^{94} + ( 4 \beta_{1} + \beta_{3} ) q^{95} + ( 4 - 4 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} + 12q^{10} + 16q^{13} - 12q^{16} + 12q^{19} - 4q^{22} - 8q^{25} + 20q^{31} + 16q^{34} - 4q^{37} - 4q^{40} + 12q^{46} + 32q^{52} - 4q^{55} + 48q^{58} + 8q^{61} - 4q^{64} + 40q^{73} + 20q^{76} - 16q^{79} - 20q^{82} + 16q^{85} - 36q^{88} - 16q^{94} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 7$$:

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

## 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
 −2.10100 −1.25928 1.25928 2.10100
−2.10100 0 2.41421 −2.10100 0 0 −0.870264 0 4.41421
1.2 −1.25928 0 −0.414214 −1.25928 0 0 3.04017 0 1.58579
1.3 1.25928 0 −0.414214 1.25928 0 0 −3.04017 0 1.58579
1.4 2.10100 0 2.41421 2.10100 0 0 0.870264 0 4.41421
 $$n$$: 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.bd yes 4
3.b odd 2 1 inner 1323.2.a.bd yes 4
7.b odd 2 1 1323.2.a.bc 4
21.c even 2 1 1323.2.a.bc 4

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

## Hecke kernels

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

 $$T_{2}^{4} - 6 T_{2}^{2} + 7$$ $$T_{5}^{4} - 6 T_{5}^{2} + 7$$ $$T_{13}^{2} - 8 T_{13} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$7 - 6 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$7 - 6 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$343 - 38 T^{2} + T^{4}$$
$13$ $$( 8 - 8 T + T^{2} )^{2}$$
$17$ $$28 - 20 T^{2} + T^{4}$$
$19$ $$( 7 - 6 T + T^{2} )^{2}$$
$23$ $$2023 - 90 T^{2} + T^{4}$$
$29$ $$1792 - 96 T^{2} + T^{4}$$
$31$ $$( -7 - 10 T + T^{2} )^{2}$$
$37$ $$( -17 + 2 T + T^{2} )^{2}$$
$41$ $$7 - 26 T^{2} + T^{4}$$
$43$ $$( -50 + T^{2} )^{2}$$
$47$ $$14812 - 244 T^{2} + T^{4}$$
$53$ $$112 - 104 T^{2} + T^{4}$$
$59$ $$28 - 12 T^{2} + T^{4}$$
$61$ $$( -68 - 4 T + T^{2} )^{2}$$
$67$ $$( -32 + T^{2} )^{2}$$
$71$ $$16807 - 294 T^{2} + T^{4}$$
$73$ $$( 98 - 20 T + T^{2} )^{2}$$
$79$ $$( -82 + 8 T + T^{2} )^{2}$$
$83$ $$112 - 216 T^{2} + T^{4}$$
$89$ $$7 - 314 T^{2} + T^{4}$$
$97$ $$( -16 - 8 T + T^{2} )^{2}$$