# Properties

 Label 5520.2.a.bt Level $5520$ Weight $2$ Character orbit 5520.a Self dual yes Analytic conductor $44.077$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} -\beta q^{11} + ( 2 + \beta ) q^{13} + q^{15} + ( -1 + \beta ) q^{17} + ( 6 - \beta ) q^{19} + ( 1 + 2 \beta ) q^{21} + q^{23} + q^{25} + q^{27} + ( -5 - \beta ) q^{29} + q^{31} -\beta q^{33} + ( 1 + 2 \beta ) q^{35} + ( 1 + 6 \beta ) q^{37} + ( 2 + \beta ) q^{39} + ( 3 - 3 \beta ) q^{41} + ( 2 - 4 \beta ) q^{43} + q^{45} + ( 2 - 3 \beta ) q^{47} + ( 2 + 4 \beta ) q^{49} + ( -1 + \beta ) q^{51} + ( 1 - 3 \beta ) q^{53} -\beta q^{55} + ( 6 - \beta ) q^{57} + ( -1 + 9 \beta ) q^{59} -\beta q^{61} + ( 1 + 2 \beta ) q^{63} + ( 2 + \beta ) q^{65} + ( 7 - 6 \beta ) q^{67} + q^{69} + ( 7 - 5 \beta ) q^{71} + ( -2 - 3 \beta ) q^{73} + q^{75} + ( -4 - \beta ) q^{77} + 12 \beta q^{79} + q^{81} + ( -7 - 3 \beta ) q^{83} + ( -1 + \beta ) q^{85} + ( -5 - \beta ) q^{87} + ( -12 - 2 \beta ) q^{89} + ( 6 + 5 \beta ) q^{91} + q^{93} + ( 6 - \beta ) q^{95} + ( 4 - 10 \beta ) q^{97} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + 4q^{13} + 2q^{15} - 2q^{17} + 12q^{19} + 2q^{21} + 2q^{23} + 2q^{25} + 2q^{27} - 10q^{29} + 2q^{31} + 2q^{35} + 2q^{37} + 4q^{39} + 6q^{41} + 4q^{43} + 2q^{45} + 4q^{47} + 4q^{49} - 2q^{51} + 2q^{53} + 12q^{57} - 2q^{59} + 2q^{63} + 4q^{65} + 14q^{67} + 2q^{69} + 14q^{71} - 4q^{73} + 2q^{75} - 8q^{77} + 2q^{81} - 14q^{83} - 2q^{85} - 10q^{87} - 24q^{89} + 12q^{91} + 2q^{93} + 12q^{95} + 8q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 1.00000 0 1.00000 0 −1.82843 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 3.82843 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.bt 2
4.b odd 2 1 2760.2.a.n 2
12.b even 2 1 8280.2.a.x 2

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

 $$T_{7}^{2} - 2 T_{7} - 7$$ $$T_{11}^{2} - 2$$ $$T_{13}^{2} - 4 T_{13} + 2$$ $$T_{17}^{2} + 2 T_{17} - 1$$ $$T_{19}^{2} - 12 T_{19} + 34$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-7 - 2 T + T^{2}$$
$11$ $$-2 + T^{2}$$
$13$ $$2 - 4 T + T^{2}$$
$17$ $$-1 + 2 T + T^{2}$$
$19$ $$34 - 12 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$23 + 10 T + T^{2}$$
$31$ $$( -1 + T )^{2}$$
$37$ $$-71 - 2 T + T^{2}$$
$41$ $$-9 - 6 T + T^{2}$$
$43$ $$-28 - 4 T + T^{2}$$
$47$ $$-14 - 4 T + T^{2}$$
$53$ $$-17 - 2 T + T^{2}$$
$59$ $$-161 + 2 T + T^{2}$$
$61$ $$-2 + T^{2}$$
$67$ $$-23 - 14 T + T^{2}$$
$71$ $$-1 - 14 T + T^{2}$$
$73$ $$-14 + 4 T + T^{2}$$
$79$ $$-288 + T^{2}$$
$83$ $$31 + 14 T + T^{2}$$
$89$ $$136 + 24 T + T^{2}$$
$97$ $$-184 - 8 T + T^{2}$$
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