# Properties

 Label 1368.2.a.j Level $1368$ Weight $2$ Character orbit 1368.a Self dual yes Analytic conductor $10.924$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
0 0 0 4.00000 0 0 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.j yes 1
3.b odd 2 1 1368.2.a.a 1
4.b odd 2 1 2736.2.a.x 1
12.b even 2 1 2736.2.a.b 1

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

 $$T_{5} - 4$$ $$T_{7}$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-4 + T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$-2 + T$$
$17$ $$-4 + T$$
$19$ $$1 + T$$
$23$ $$-6 + T$$
$29$ $$-10 + T$$
$31$ $$-8 + T$$
$37$ $$10 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-6 + T$$
$53$ $$-2 + T$$
$59$ $$-4 + T$$
$61$ $$-10 + T$$
$67$ $$12 + T$$
$71$ $$-12 + T$$
$73$ $$6 + T$$
$79$ $$4 + T$$
$83$ $$-14 + T$$
$89$ $$6 + T$$
$97$ $$2 + T$$
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