# Properties

 Label 4368.2.a.s Level $4368$ Weight $2$ Character orbit 4368.a Self dual yes Analytic conductor $34.879$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.8786556029$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} + q^{7} + q^{9} + q^{11} + q^{13} - q^{15} - q^{17} - 7q^{19} + q^{21} - 3q^{23} - 4q^{25} + q^{27} - 3q^{29} - 8q^{31} + q^{33} - q^{35} + 7q^{37} + q^{39} + 8q^{41} - 7q^{43} - q^{45} - 8q^{47} + q^{49} - q^{51} - 10q^{53} - q^{55} - 7q^{57} - 4q^{59} + 7q^{61} + q^{63} - q^{65} - 2q^{67} - 3q^{69} - 4q^{71} - q^{73} - 4q^{75} + q^{77} - 2q^{79} + q^{81} + 6q^{83} + q^{85} - 3q^{87} + 14q^{89} + q^{91} - 8q^{93} + 7q^{95} - 14q^{97} + q^{99} + O(q^{100})$$

## 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 1.00000 0 −1.00000 0 1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 4368.2.a.s 1
4.b odd 2 1 546.2.a.a 1
12.b even 2 1 1638.2.a.r 1
28.d even 2 1 3822.2.a.n 1
52.b odd 2 1 7098.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.a.a 1 4.b odd 2 1
1638.2.a.r 1 12.b even 2 1
3822.2.a.n 1 28.d even 2 1
4368.2.a.s 1 1.a even 1 1 trivial
7098.2.a.t 1 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4368))$$:

 $$T_{5} + 1$$ $$T_{11} - 1$$ $$T_{17} + 1$$

## Hecke characteristic polynomials

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