# Properties

 Label 5520.2.a.bz 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: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2760) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{1} + \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} - q^{15} + ( -1 - \beta_{1} ) q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{21} + q^{23} + q^{25} - q^{27} + ( 7 - \beta_{1} ) q^{29} + ( -5 - 2 \beta_{1} - 3 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - \beta_{2} ) q^{33} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{35} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -\beta_{1} + \beta_{2} ) q^{39} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 - 2 \beta_{2} ) q^{43} + q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} + ( 8 + 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( 1 + \beta_{1} ) q^{51} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{53} + ( -2 - \beta_{1} + \beta_{2} ) q^{55} + ( \beta_{1} + \beta_{2} ) q^{57} + ( -5 - 3 \beta_{1} - 4 \beta_{2} ) q^{59} + ( -6 - \beta_{1} + 3 \beta_{2} ) q^{61} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} ) q^{65} + ( -3 - 2 \beta_{1} - 5 \beta_{2} ) q^{67} - q^{69} + ( 3 - 5 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -4 + 5 \beta_{1} + 3 \beta_{2} ) q^{73} - q^{75} + ( -2 - 9 \beta_{1} - 7 \beta_{2} ) q^{77} -4 \beta_{1} q^{79} + q^{81} + ( -7 - \beta_{1} ) q^{83} + ( -1 - \beta_{1} ) q^{85} + ( -7 + \beta_{1} ) q^{87} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 4 + 5 \beta_{1} + 5 \beta_{2} ) q^{91} + ( 5 + 2 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -\beta_{1} - \beta_{2} ) q^{95} + ( 6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( -2 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} - 2q^{7} + 3q^{9} - 8q^{11} + 2q^{13} - 3q^{15} - 4q^{17} + 2q^{21} + 3q^{23} + 3q^{25} - 3q^{27} + 20q^{29} - 14q^{31} + 8q^{33} - 2q^{35} - 4q^{37} - 2q^{39} - 8q^{41} + 8q^{43} + 3q^{45} - 12q^{47} + 29q^{49} + 4q^{51} - 8q^{55} - 14q^{59} - 22q^{61} - 2q^{63} + 2q^{65} - 6q^{67} - 3q^{69} + 6q^{71} - 10q^{73} - 3q^{75} - 8q^{77} - 4q^{79} + 3q^{81} - 22q^{83} - 4q^{85} - 20q^{87} - 10q^{89} + 12q^{91} + 14q^{93} + 2q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −0.363328 −1.76156 3.12489
0 −1.00000 0 1.00000 0 −4.86799 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.89692 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 4.76491 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.bz 3
4.b odd 2 1 2760.2.a.u 3
12.b even 2 1 8280.2.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.a.u 3 4.b odd 2 1
5520.2.a.bz 3 1.a even 1 1 trivial
8280.2.a.bi 3 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 23 T_{7} - 44$$ $$T_{11}^{3} + 8 T_{11}^{2} + 2 T_{11} - 64$$ $$T_{13}^{3} - 2 T_{13}^{2} - 18 T_{13} + 44$$ $$T_{17}^{3} + 4 T_{17}^{2} - T_{17} - 2$$ $$T_{19}^{3} - 10 T_{19} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$-44 - 23 T + 2 T^{2} + T^{3}$$
$11$ $$-64 + 2 T + 8 T^{2} + T^{3}$$
$13$ $$44 - 18 T - 2 T^{2} + T^{3}$$
$17$ $$-2 - T + 4 T^{2} + T^{3}$$
$19$ $$-8 - 10 T + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$-250 + 127 T - 20 T^{2} + T^{3}$$
$31$ $$-472 - 7 T + 14 T^{2} + T^{3}$$
$37$ $$-2 - 19 T + 4 T^{2} + T^{3}$$
$41$ $$-670 - 97 T + 8 T^{2} + T^{3}$$
$43$ $$80 - 12 T - 8 T^{2} + T^{3}$$
$47$ $$32 + 38 T + 12 T^{2} + T^{3}$$
$53$ $$-122 - 49 T + T^{3}$$
$59$ $$-1100 - 69 T + 14 T^{2} + T^{3}$$
$61$ $$-580 + 66 T + 22 T^{2} + T^{3}$$
$67$ $$-1156 - 175 T + 6 T^{2} + T^{3}$$
$71$ $$848 - 133 T - 6 T^{2} + T^{3}$$
$73$ $$-764 - 130 T + 10 T^{2} + T^{3}$$
$79$ $$128 - 96 T + 4 T^{2} + T^{3}$$
$83$ $$352 + 155 T + 22 T^{2} + T^{3}$$
$89$ $$-848 - 88 T + 10 T^{2} + T^{3}$$
$97$ $$-16 - 248 T - 2 T^{2} + T^{3}$$