# Properties

 Label 5040.2.a.by Level $5040$ Weight $2$ Character orbit 5040.a Self dual yes Analytic conductor $40.245$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5040.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.2446026187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} +O(q^{10})$$ $$q + q^{5} + q^{7} + ( 4 - \beta ) q^{11} + ( 2 - \beta ) q^{13} + ( -2 - \beta ) q^{17} -2 \beta q^{19} + 2 \beta q^{23} + q^{25} + ( 2 - \beta ) q^{29} + 8 q^{31} + q^{35} -2 q^{37} + ( -2 + 2 \beta ) q^{41} + ( 4 - 2 \beta ) q^{43} + 3 \beta q^{47} + q^{49} + ( -6 + 2 \beta ) q^{53} + ( 4 - \beta ) q^{55} + 8 q^{59} + ( 2 + 2 \beta ) q^{61} + ( 2 - \beta ) q^{65} + 4 q^{67} + 8 q^{71} -6 q^{73} + ( 4 - \beta ) q^{77} + ( -8 + 3 \beta ) q^{79} + 4 \beta q^{83} + ( -2 - \beta ) q^{85} + ( -10 + 2 \beta ) q^{89} + ( 2 - \beta ) q^{91} -2 \beta q^{95} + ( 2 + 5 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + 2q^{7} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{7} + 7q^{11} + 3q^{13} - 5q^{17} - 2q^{19} + 2q^{23} + 2q^{25} + 3q^{29} + 16q^{31} + 2q^{35} - 4q^{37} - 2q^{41} + 6q^{43} + 3q^{47} + 2q^{49} - 10q^{53} + 7q^{55} + 16q^{59} + 6q^{61} + 3q^{65} + 8q^{67} + 16q^{71} - 12q^{73} + 7q^{77} - 13q^{79} + 4q^{83} - 5q^{85} - 18q^{89} + 3q^{91} - 2q^{95} + 9q^{97} + O(q^{100})$$

## 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
 3.37228 −2.37228
0 0 0 1.00000 0 1.00000 0 0 0
1.2 0 0 0 1.00000 0 1.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 5040.2.a.by 2
3.b odd 2 1 560.2.a.h 2
4.b odd 2 1 2520.2.a.x 2
12.b even 2 1 280.2.a.c 2
15.d odd 2 1 2800.2.a.bk 2
15.e even 4 2 2800.2.g.r 4
21.c even 2 1 3920.2.a.bt 2
24.f even 2 1 2240.2.a.bk 2
24.h odd 2 1 2240.2.a.bg 2
60.h even 2 1 1400.2.a.r 2
60.l odd 4 2 1400.2.g.i 4
84.h odd 2 1 1960.2.a.s 2
84.j odd 6 2 1960.2.q.r 4
84.n even 6 2 1960.2.q.t 4
420.o odd 2 1 9800.2.a.bu 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 12.b even 2 1
560.2.a.h 2 3.b odd 2 1
1400.2.a.r 2 60.h even 2 1
1400.2.g.i 4 60.l odd 4 2
1960.2.a.s 2 84.h odd 2 1
1960.2.q.r 4 84.j odd 6 2
1960.2.q.t 4 84.n even 6 2
2240.2.a.bg 2 24.h odd 2 1
2240.2.a.bk 2 24.f even 2 1
2520.2.a.x 2 4.b odd 2 1
2800.2.a.bk 2 15.d odd 2 1
2800.2.g.r 4 15.e even 4 2
3920.2.a.bt 2 21.c even 2 1
5040.2.a.by 2 1.a even 1 1 trivial
9800.2.a.bu 2 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5040))$$:

 $$T_{11}^{2} - 7 T_{11} + 4$$ $$T_{13}^{2} - 3 T_{13} - 6$$ $$T_{17}^{2} + 5 T_{17} - 2$$ $$T_{19}^{2} + 2 T_{19} - 32$$ $$T_{29}^{2} - 3 T_{29} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$4 - 7 T + T^{2}$$
$13$ $$-6 - 3 T + T^{2}$$
$17$ $$-2 + 5 T + T^{2}$$
$19$ $$-32 + 2 T + T^{2}$$
$23$ $$-32 - 2 T + T^{2}$$
$29$ $$-6 - 3 T + T^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-32 + 2 T + T^{2}$$
$43$ $$-24 - 6 T + T^{2}$$
$47$ $$-72 - 3 T + T^{2}$$
$53$ $$-8 + 10 T + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$-24 - 6 T + T^{2}$$
$67$ $$( -4 + T )^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$-32 + 13 T + T^{2}$$
$83$ $$-128 - 4 T + T^{2}$$
$89$ $$48 + 18 T + T^{2}$$
$97$ $$-186 - 9 T + T^{2}$$