# Properties

 Label 1368.2.a.o Level $1368$ Weight $2$ Character orbit 1368.a Self dual yes Analytic conductor $10.924$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1368 = 2^{3} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1368.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.9235349965$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 Defining polynomial: $$x^{3} - x^{2} - 8 x + 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} ) q^{7} + ( -3 - \beta_{1} ) q^{11} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{17} - q^{19} + ( -2 + 2 \beta_{1} ) q^{23} + ( 2 - 3 \beta_{1} + \beta_{2} ) q^{25} + ( -2 + 2 \beta_{2} ) q^{29} -6 q^{31} + ( -9 + 3 \beta_{1} - 2 \beta_{2} ) q^{35} + 4 q^{37} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -3 - \beta_{1} - \beta_{2} ) q^{43} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{47} + ( 10 - \beta_{1} + 3 \beta_{2} ) q^{49} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 + \beta_{1} + \beta_{2} ) q^{55} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{61} + ( -10 + 6 \beta_{1} ) q^{65} -4 \beta_{2} q^{67} -4 \beta_{2} q^{71} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{73} + ( -5 - \beta_{1} - 6 \beta_{2} ) q^{77} + ( -8 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -6 + 2 \beta_{2} ) q^{83} + ( -5 + 5 \beta_{1} + \beta_{2} ) q^{85} + ( -2 - 2 \beta_{2} ) q^{89} + ( 2 - 10 \beta_{1} + 2 \beta_{2} ) q^{91} + ( -1 + \beta_{1} ) q^{95} + ( 6 + 4 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{5} - 2 q^{7} + O(q^{10})$$ $$3 q + 2 q^{5} - 2 q^{7} - 10 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} - 4 q^{23} + 3 q^{25} - 6 q^{29} - 18 q^{31} - 24 q^{35} + 12 q^{37} - 14 q^{41} - 10 q^{43} - 8 q^{47} + 29 q^{49} - 10 q^{53} + 10 q^{55} + 8 q^{59} - 24 q^{65} + 16 q^{73} - 16 q^{77} - 22 q^{79} - 18 q^{83} - 10 q^{85} - 6 q^{89} - 4 q^{91} - 2 q^{95} + 22 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8 x + 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta_{1} + 6$$

## 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
 2.59774 1.31955 −2.91729
0 0 0 −1.59774 0 4.94370 0 0 0
1.2 0 0 0 −0.319551 0 −2.61968 0 0 0
1.3 0 0 0 3.91729 0 −4.32401 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$19$$ $$1$$

## 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 1368.2.a.o yes 3
3.b odd 2 1 1368.2.a.m 3
4.b odd 2 1 2736.2.a.be 3
12.b even 2 1 2736.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.a.m 3 3.b odd 2 1
1368.2.a.o yes 3 1.a even 1 1 trivial
2736.2.a.bc 3 12.b even 2 1
2736.2.a.be 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1368))$$:

 $$T_{5}^{3} - 2 T_{5}^{2} - 7 T_{5} - 2$$ $$T_{7}^{3} + 2 T_{7}^{2} - 23 T_{7} - 56$$ $$T_{11}^{3} + 10 T_{11}^{2} + 25 T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$-2 - 7 T - 2 T^{2} + T^{3}$$
$7$ $$-56 - 23 T + 2 T^{2} + T^{3}$$
$11$ $$2 + 25 T + 10 T^{2} + T^{3}$$
$13$ $$-160 - 44 T + 4 T^{2} + T^{3}$$
$17$ $$-152 - 15 T + 8 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$16 - 28 T + 4 T^{2} + T^{3}$$
$29$ $$-104 - 28 T + 6 T^{2} + T^{3}$$
$31$ $$( 6 + T )^{3}$$
$37$ $$( -4 + T )^{3}$$
$41$ $$-32 + 16 T + 14 T^{2} + T^{3}$$
$43$ $$-4 + 9 T + 10 T^{2} + T^{3}$$
$47$ $$-320 - 39 T + 8 T^{2} + T^{3}$$
$53$ $$-664 - 92 T + 10 T^{2} + T^{3}$$
$59$ $$1280 - 176 T - 8 T^{2} + T^{3}$$
$61$ $$-190 - 67 T + T^{3}$$
$67$ $$256 - 160 T + T^{3}$$
$71$ $$256 - 160 T + T^{3}$$
$73$ $$122 + 5 T - 16 T^{2} + T^{3}$$
$79$ $$-64 + 112 T + 22 T^{2} + T^{3}$$
$83$ $$-56 + 68 T + 18 T^{2} + T^{3}$$
$89$ $$-40 - 28 T + 6 T^{2} + T^{3}$$
$97$ $$1048 + 28 T - 22 T^{2} + T^{3}$$