# Properties

 Label 1323.2.a.y Level $1323$ Weight $2$ Character orbit 1323.a Self dual yes Analytic conductor $10.564$ Analytic rank $1$ Dimension $3$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} -\beta_{2} q^{5} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{8} + ( 1 - \beta_{1} + \beta_{2} ) q^{10} + ( -2 - \beta_{1} ) q^{11} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{13} + ( 5 - 4 \beta_{1} + \beta_{2} ) q^{16} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( -1 - \beta_{1} + \beta_{2} ) q^{19} + ( -5 + 2 \beta_{1} ) q^{20} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{22} + ( -2 - \beta_{1} - \beta_{2} ) q^{23} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{25} + ( 6 + \beta_{2} ) q^{26} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( -10 + 4 \beta_{1} - \beta_{2} ) q^{32} + ( -6 + 3 \beta_{1} - 3 \beta_{2} ) q^{34} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{37} + ( -3 - 2 \beta_{2} ) q^{38} + ( 9 - 3 \beta_{1} ) q^{40} + ( 2 - 5 \beta_{1} - \beta_{2} ) q^{41} + ( -4 + 3 \beta_{1} ) q^{43} -\beta_{2} q^{44} -3 \beta_{1} q^{46} + ( 2 + 4 \beta_{1} + \beta_{2} ) q^{47} + ( -3 \beta_{1} + \beta_{2} ) q^{50} + ( -5 + 3 \beta_{1} - 3 \beta_{2} ) q^{52} + ( -7 - 2 \beta_{1} + \beta_{2} ) q^{53} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{55} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{58} + ( -5 - \beta_{1} - \beta_{2} ) q^{59} + ( -1 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -8 + 2 \beta_{1} - \beta_{2} ) q^{62} + ( 13 - 3 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{65} + ( 4 + \beta_{1} - \beta_{2} ) q^{67} + ( 16 - 7 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{71} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 - 5 \beta_{1} + \beta_{2} ) q^{74} + ( 7 - 3 \beta_{1} ) q^{76} + ( 2 + 3 \beta_{1} + 3 \beta_{2} ) q^{79} + ( -8 + 5 \beta_{1} - 3 \beta_{2} ) q^{80} + ( -16 + \beta_{1} - 4 \beta_{2} ) q^{82} + ( 6 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -9 + 3 \beta_{1} + 3 \beta_{2} ) q^{85} + ( 13 - 4 \beta_{1} + 3 \beta_{2} ) q^{86} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{88} + ( -3 - 4 \beta_{2} ) q^{89} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{92} + ( 9 + 3 \beta_{1} + 3 \beta_{2} ) q^{94} + ( -5 + 2 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 6 - 2 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{2} + 4q^{4} + q^{5} - 9q^{8} + O(q^{10})$$ $$3q - 2q^{2} + 4q^{4} + q^{5} - 9q^{8} + q^{10} - 7q^{11} - 2q^{13} + 10q^{16} - 5q^{19} - 13q^{20} - 4q^{22} - 6q^{23} - 2q^{25} + 17q^{26} - 13q^{29} + 8q^{31} - 25q^{32} - 12q^{34} - 8q^{37} - 7q^{38} + 24q^{40} + 2q^{41} - 9q^{43} + q^{44} - 3q^{46} + 9q^{47} - 4q^{50} - 9q^{52} - 24q^{53} - 4q^{55} + 14q^{58} - 15q^{59} + q^{61} - 21q^{62} + 33q^{64} - 10q^{65} + 14q^{67} + 39q^{68} + 3q^{71} - 7q^{73} + 18q^{76} + 6q^{79} - 16q^{80} - 43q^{82} + 3q^{83} - 27q^{85} + 32q^{86} + 9q^{88} - 5q^{89} - 12q^{92} + 27q^{94} - 16q^{95} + 14q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

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

## 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
 −1.69963 0.239123 2.46050
−2.69963 0 5.28799 −1.58836 0 0 −8.87636 0 4.28799
1.2 −0.760877 0 −1.42107 3.18194 0 0 2.60301 0 −2.42107
1.3 1.46050 0 0.133074 −0.593579 0 0 −2.72665 0 −0.866926
 $$n$$: 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.y 3
3.b odd 2 1 1323.2.a.z 3
7.b odd 2 1 1323.2.a.x 3
7.d odd 6 2 189.2.e.f yes 6
21.c even 2 1 1323.2.a.ba 3
21.g even 6 2 189.2.e.e 6
63.i even 6 2 567.2.h.i 6
63.k odd 6 2 567.2.g.i 6
63.s even 6 2 567.2.g.h 6
63.t odd 6 2 567.2.h.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e 6 21.g even 6 2
189.2.e.f yes 6 7.d odd 6 2
567.2.g.h 6 63.s even 6 2
567.2.g.i 6 63.k odd 6 2
567.2.h.h 6 63.t odd 6 2
567.2.h.i 6 63.i even 6 2
1323.2.a.x 3 7.b odd 2 1
1323.2.a.y 3 1.a even 1 1 trivial
1323.2.a.z 3 3.b odd 2 1
1323.2.a.ba 3 21.c even 2 1

## Hecke kernels

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

 $$T_{2}^{3} + 2 T_{2}^{2} - 3 T_{2} - 3$$ $$T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 3$$ $$T_{13}^{3} + 2 T_{13}^{2} - 19 T_{13} - 47$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 - 3 T + 2 T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-3 - 6 T - T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$3 + 12 T + 7 T^{2} + T^{3}$$
$13$ $$-47 - 19 T + 2 T^{2} + T^{3}$$
$17$ $$-9 - 33 T + T^{3}$$
$19$ $$-29 - 4 T + 5 T^{2} + T^{3}$$
$23$ $$-9 + 3 T + 6 T^{2} + T^{3}$$
$29$ $$9 + 30 T + 13 T^{2} + T^{3}$$
$31$ $$69 + T - 8 T^{2} + T^{3}$$
$37$ $$-93 - 5 T + 8 T^{2} + T^{3}$$
$41$ $$387 - 105 T - 2 T^{2} + T^{3}$$
$43$ $$-101 - 12 T + 9 T^{2} + T^{3}$$
$47$ $$-9 - 42 T - 9 T^{2} + T^{3}$$
$53$ $$243 + 165 T + 24 T^{2} + T^{3}$$
$59$ $$81 + 66 T + 15 T^{2} + T^{3}$$
$61$ $$121 - 49 T - T^{2} + T^{3}$$
$67$ $$-31 + 53 T - 14 T^{2} + T^{3}$$
$71$ $$-243 - 108 T - 3 T^{2} + T^{3}$$
$73$ $$-981 - 134 T + 7 T^{2} + T^{3}$$
$79$ $$127 - 69 T - 6 T^{2} + T^{3}$$
$83$ $$-729 - 180 T - 3 T^{2} + T^{3}$$
$89$ $$-489 - 93 T + 5 T^{2} + T^{3}$$
$97$ $$24 + 16 T - 14 T^{2} + T^{3}$$