# Properties

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

# 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: $$0$$ 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 345) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + ( 1 + 2 \beta ) q^{7} + q^{9} + ( 4 + \beta ) q^{11} + ( 2 - \beta ) q^{13} - q^{15} + ( 1 + 5 \beta ) q^{17} + ( 2 - 3 \beta ) q^{19} + ( 1 + 2 \beta ) q^{21} + q^{23} + q^{25} + q^{27} + ( -5 - \beta ) q^{29} + ( 7 + 2 \beta ) q^{31} + ( 4 + \beta ) q^{33} + ( -1 - 2 \beta ) q^{35} + 3 q^{37} + ( 2 - \beta ) q^{39} + ( -9 + \beta ) q^{41} -6 q^{43} - q^{45} + ( 10 + \beta ) q^{47} + ( 2 + 4 \beta ) q^{49} + ( 1 + 5 \beta ) q^{51} + ( 3 - 3 \beta ) q^{53} + ( -4 - \beta ) q^{55} + ( 2 - 3 \beta ) q^{57} + ( -3 - 7 \beta ) q^{59} + ( 4 - \beta ) q^{61} + ( 1 + 2 \beta ) q^{63} + ( -2 + \beta ) q^{65} + ( 3 - 6 \beta ) q^{67} + q^{69} + ( 1 + 7 \beta ) q^{71} + ( -6 + 3 \beta ) q^{73} + q^{75} + ( 8 + 9 \beta ) q^{77} + ( 4 - 8 \beta ) q^{79} + q^{81} + ( -7 + 5 \beta ) q^{83} + ( -1 - 5 \beta ) q^{85} + ( -5 - \beta ) q^{87} + ( -4 - 10 \beta ) q^{89} + ( -2 + 3 \beta ) q^{91} + ( 7 + 2 \beta ) q^{93} + ( -2 + 3 \beta ) q^{95} + ( -4 - 2 \beta ) q^{97} + ( 4 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{5} + 2q^{7} + 2q^{9} + 8q^{11} + 4q^{13} - 2q^{15} + 2q^{17} + 4q^{19} + 2q^{21} + 2q^{23} + 2q^{25} + 2q^{27} - 10q^{29} + 14q^{31} + 8q^{33} - 2q^{35} + 6q^{37} + 4q^{39} - 18q^{41} - 12q^{43} - 2q^{45} + 20q^{47} + 4q^{49} + 2q^{51} + 6q^{53} - 8q^{55} + 4q^{57} - 6q^{59} + 8q^{61} + 2q^{63} - 4q^{65} + 6q^{67} + 2q^{69} + 2q^{71} - 12q^{73} + 2q^{75} + 16q^{77} + 8q^{79} + 2q^{81} - 14q^{83} - 2q^{85} - 10q^{87} - 8q^{89} - 4q^{91} + 14q^{93} - 4q^{95} - 8q^{97} + 8q^{99} + O(q^{100})$$

## 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
 −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.bm 2
4.b odd 2 1 345.2.a.h 2
12.b even 2 1 1035.2.a.j 2
20.d odd 2 1 1725.2.a.z 2
20.e even 4 2 1725.2.b.s 4
60.h even 2 1 5175.2.a.bj 2
92.b even 2 1 7935.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.h 2 4.b odd 2 1
1035.2.a.j 2 12.b even 2 1
1725.2.a.z 2 20.d odd 2 1
1725.2.b.s 4 20.e even 4 2
5175.2.a.bj 2 60.h even 2 1
5520.2.a.bm 2 1.a even 1 1 trivial
7935.2.a.q 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5520))$$:

 $$T_{7}^{2} - 2 T_{7} - 7$$ $$T_{11}^{2} - 8 T_{11} + 14$$ $$T_{13}^{2} - 4 T_{13} + 2$$ $$T_{17}^{2} - 2 T_{17} - 49$$ $$T_{19}^{2} - 4 T_{19} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$-7 - 2 T + T^{2}$$
$11$ $$14 - 8 T + T^{2}$$
$13$ $$2 - 4 T + T^{2}$$
$17$ $$-49 - 2 T + T^{2}$$
$19$ $$-14 - 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$23 + 10 T + T^{2}$$
$31$ $$41 - 14 T + T^{2}$$
$37$ $$( -3 + T )^{2}$$
$41$ $$79 + 18 T + T^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$98 - 20 T + T^{2}$$
$53$ $$-9 - 6 T + T^{2}$$
$59$ $$-89 + 6 T + T^{2}$$
$61$ $$14 - 8 T + T^{2}$$
$67$ $$-63 - 6 T + T^{2}$$
$71$ $$-97 - 2 T + T^{2}$$
$73$ $$18 + 12 T + T^{2}$$
$79$ $$-112 - 8 T + T^{2}$$
$83$ $$-1 + 14 T + T^{2}$$
$89$ $$-184 + 8 T + T^{2}$$
$97$ $$8 + 8 T + T^{2}$$