# Properties

 Label 5390.2.a.z Level $5390$ Weight $2$ Character orbit 5390.a Self dual yes Analytic conductor $43.039$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5390.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.0393666895$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} - 2 q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} - 2 q^{9} + q^{10} + q^{11} - q^{12} + q^{13} - q^{15} + q^{16} - 2 q^{18} - 2 q^{19} + q^{20} + q^{22} - 6 q^{23} - q^{24} + q^{25} + q^{26} + 5 q^{27} - 9 q^{29} - q^{30} + 4 q^{31} + q^{32} - q^{33} - 2 q^{36} - 4 q^{37} - 2 q^{38} - q^{39} + q^{40} - 12 q^{41} - 10 q^{43} + q^{44} - 2 q^{45} - 6 q^{46} + 12 q^{47} - q^{48} + q^{50} + q^{52} + 6 q^{53} + 5 q^{54} + q^{55} + 2 q^{57} - 9 q^{58} - 3 q^{59} - q^{60} + 7 q^{61} + 4 q^{62} + q^{64} + q^{65} - q^{66} - 13 q^{67} + 6 q^{69} - 2 q^{72} - 14 q^{73} - 4 q^{74} - q^{75} - 2 q^{76} - q^{78} + 17 q^{79} + q^{80} + q^{81} - 12 q^{82} - 6 q^{83} - 10 q^{86} + 9 q^{87} + q^{88} + 6 q^{89} - 2 q^{90} - 6 q^{92} - 4 q^{93} + 12 q^{94} - 2 q^{95} - q^{96} - 5 q^{97} - 2 q^{99} + O(q^{100})$$

## 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 0 1.00000 −2.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 5390.2.a.z 1
7.b odd 2 1 5390.2.a.be 1
7.d odd 6 2 770.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.a 2 7.d odd 6 2
5390.2.a.z 1 1.a even 1 1 trivial
5390.2.a.be 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5390))$$:

 $$T_{3} + 1$$ $$T_{13} - 1$$ $$T_{17}$$ $$T_{19} + 2$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$-1 + T$$
$17$ $$T$$
$19$ $$2 + T$$
$23$ $$6 + T$$
$29$ $$9 + T$$
$31$ $$-4 + T$$
$37$ $$4 + T$$
$41$ $$12 + T$$
$43$ $$10 + T$$
$47$ $$-12 + T$$
$53$ $$-6 + T$$
$59$ $$3 + T$$
$61$ $$-7 + T$$
$67$ $$13 + T$$
$71$ $$T$$
$73$ $$14 + T$$
$79$ $$-17 + T$$
$83$ $$6 + T$$
$89$ $$-6 + T$$
$97$ $$5 + T$$