# Properties

 Label 5520.2.a.x Level $5520$ Weight $2$ Character orbit 5520.a Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + q^{9} + O(q^{10})$$ $$q + q^{3} - q^{5} + q^{9} + 6q^{13} - q^{15} + 2q^{17} + q^{23} + q^{25} + q^{27} + 6q^{29} - 8q^{31} + 10q^{37} + 6q^{39} - 6q^{41} + 8q^{43} - q^{45} - 8q^{47} - 7q^{49} + 2q^{51} - 6q^{53} + 4q^{59} - 6q^{61} - 6q^{65} - 8q^{67} + q^{69} + 8q^{71} + 10q^{73} + q^{75} + 8q^{79} + q^{81} + 8q^{83} - 2q^{85} + 6q^{87} - 6q^{89} - 8q^{93} + 18q^{97} + O(q^{100})$$

## 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 1.00000 0 −1.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 5520.2.a.x 1
4.b odd 2 1 690.2.a.h 1
12.b even 2 1 2070.2.a.h 1
20.d odd 2 1 3450.2.a.j 1
20.e even 4 2 3450.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.h 1 4.b odd 2 1
2070.2.a.h 1 12.b even 2 1
3450.2.a.j 1 20.d odd 2 1
3450.2.d.e 2 20.e even 4 2
5520.2.a.x 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}$$ $$T_{11}$$ $$T_{13} - 6$$ $$T_{17} - 2$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$-2 + T$$
$19$ $$T$$
$23$ $$-1 + T$$
$29$ $$-6 + T$$
$31$ $$8 + T$$
$37$ $$-10 + T$$
$41$ $$6 + T$$
$43$ $$-8 + T$$
$47$ $$8 + T$$
$53$ $$6 + T$$
$59$ $$-4 + T$$
$61$ $$6 + T$$
$67$ $$8 + T$$
$71$ $$-8 + T$$
$73$ $$-10 + T$$
$79$ $$-8 + T$$
$83$ $$-8 + T$$
$89$ $$6 + T$$
$97$ $$-18 + T$$
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