# Properties

 Label 5520.2.a.by Level $5520$ Weight $2$ Character orbit 5520.a Self dual yes Analytic conductor $44.077$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + ( -2 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} + ( -2 - \beta_{2} ) q^{7} + q^{9} + ( 1 - \beta_{1} + \beta_{2} ) q^{11} + ( 1 + \beta_{1} - \beta_{2} ) q^{13} - q^{15} + ( -1 + \beta_{1} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} ) q^{19} + ( 2 + \beta_{2} ) q^{21} - q^{23} + q^{25} - q^{27} + ( -1 - 3 \beta_{1} ) q^{29} + ( -2 \beta_{1} + \beta_{2} ) q^{31} + ( -1 + \beta_{1} - \beta_{2} ) q^{33} + ( -2 - \beta_{2} ) q^{35} -\beta_{2} q^{37} + ( -1 - \beta_{1} + \beta_{2} ) q^{39} + ( -1 + \beta_{1} + 4 \beta_{2} ) q^{41} + ( -6 - 2 \beta_{1} ) q^{43} + q^{45} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{47} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{49} + ( 1 - \beta_{1} ) q^{51} + ( -3 + 3 \beta_{1} ) q^{53} + ( 1 - \beta_{1} + \beta_{2} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} ) q^{57} + ( 3 + 3 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -1 - \beta_{1} - \beta_{2} ) q^{61} + ( -2 - \beta_{2} ) q^{63} + ( 1 + \beta_{1} - \beta_{2} ) q^{65} + ( 6 \beta_{1} - \beta_{2} ) q^{67} + q^{69} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{71} + ( -3 + \beta_{1} - 5 \beta_{2} ) q^{73} - q^{75} + ( -5 + \beta_{1} - \beta_{2} ) q^{77} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{79} + q^{81} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 + \beta_{1} ) q^{85} + ( 1 + 3 \beta_{1} ) q^{87} + ( 4 - 6 \beta_{1} + 4 \beta_{2} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} ) q^{91} + ( 2 \beta_{1} - \beta_{2} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} ) q^{95} + ( 6 - 4 \beta_{1} + 6 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} - 6q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} - 6q^{7} + 3q^{9} + 2q^{11} + 4q^{13} - 3q^{15} - 2q^{17} - 2q^{19} + 6q^{21} - 3q^{23} + 3q^{25} - 3q^{27} - 6q^{29} - 2q^{31} - 2q^{33} - 6q^{35} - 4q^{39} - 2q^{41} - 20q^{43} + 3q^{45} + 6q^{47} + 5q^{49} + 2q^{51} - 6q^{53} + 2q^{55} + 2q^{57} + 12q^{59} - 4q^{61} - 6q^{63} + 4q^{65} + 6q^{67} + 3q^{69} + 8q^{71} - 8q^{73} - 3q^{75} - 14q^{77} + 4q^{79} + 3q^{81} - 8q^{83} - 2q^{85} + 6q^{87} + 6q^{89} + 2q^{91} + 2q^{93} - 2q^{95} + 14q^{97} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 2.34292 −1.81361 0.470683
0 −1.00000 0 1.00000 0 −4.48929 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −2.28917 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 0.778457 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.by 3
4.b odd 2 1 345.2.a.j 3
12.b even 2 1 1035.2.a.n 3
20.d odd 2 1 1725.2.a.bi 3
20.e even 4 2 1725.2.b.u 6
60.h even 2 1 5175.2.a.br 3
92.b even 2 1 7935.2.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.j 3 4.b odd 2 1
1035.2.a.n 3 12.b even 2 1
1725.2.a.bi 3 20.d odd 2 1
1725.2.b.u 6 20.e even 4 2
5175.2.a.br 3 60.h even 2 1
5520.2.a.by 3 1.a even 1 1 trivial
7935.2.a.u 3 92.b even 2 1

## Hecke kernels

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

 $$T_{7}^{3} + 6 T_{7}^{2} + 5 T_{7} - 8$$ $$T_{11}^{3} - 2 T_{11}^{2} - 6 T_{11} + 8$$ $$T_{13}^{3} - 4 T_{13}^{2} - 2 T_{13} + 4$$ $$T_{17}^{3} + 2 T_{17}^{2} - 3 T_{17} - 2$$ $$T_{19}^{3} + 2 T_{19}^{2} - 14 T_{19} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-8 + 5 T + 6 T^{2} + T^{3}$$
$11$ $$8 - 6 T - 2 T^{2} + T^{3}$$
$13$ $$4 - 2 T - 4 T^{2} + T^{3}$$
$17$ $$-2 - 3 T + 2 T^{2} + T^{3}$$
$19$ $$-32 - 14 T + 2 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-86 - 27 T + 6 T^{2} + T^{3}$$
$31$ $$-32 - 15 T + 2 T^{2} + T^{3}$$
$37$ $$-2 - 7 T + T^{3}$$
$41$ $$-218 - 131 T + 2 T^{2} + T^{3}$$
$43$ $$176 + 116 T + 20 T^{2} + T^{3}$$
$47$ $$-16 - 22 T - 6 T^{2} + T^{3}$$
$53$ $$-54 - 27 T + 6 T^{2} + T^{3}$$
$59$ $$-32 - 43 T - 12 T^{2} + T^{3}$$
$61$ $$4 - 10 T + 4 T^{2} + T^{3}$$
$67$ $$724 - 127 T - 6 T^{2} + T^{3}$$
$71$ $$148 - 19 T - 8 T^{2} + T^{3}$$
$73$ $$-932 - 138 T + 8 T^{2} + T^{3}$$
$79$ $$-64 - 112 T - 4 T^{2} + T^{3}$$
$83$ $$-92 - 3 T + 8 T^{2} + T^{3}$$
$89$ $$-16 - 160 T - 6 T^{2} + T^{3}$$
$97$ $$2176 - 160 T - 14 T^{2} + T^{3}$$