# Properties

 Label 5040.2.a.bm Level $5040$ Weight $2$ Character orbit 5040.a Self dual yes Analytic conductor $40.245$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} + O(q^{10})$$ $$q + q^{5} + q^{7} + 4q^{11} - 6q^{13} - 2q^{17} + q^{25} - 6q^{29} - 8q^{31} + q^{35} - 10q^{37} - 2q^{41} - 4q^{43} + 8q^{47} + q^{49} + 2q^{53} + 4q^{55} - 8q^{59} - 14q^{61} - 6q^{65} + 12q^{67} - 16q^{71} + 2q^{73} + 4q^{77} + 8q^{79} + 8q^{83} - 2q^{85} - 10q^{89} - 6q^{91} + 2q^{97} + O(q^{100})$$

## 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
 0
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.bm 1
3.b odd 2 1 560.2.a.d 1
4.b odd 2 1 630.2.a.d 1
12.b even 2 1 70.2.a.a 1
15.d odd 2 1 2800.2.a.m 1
15.e even 4 2 2800.2.g.n 2
20.d odd 2 1 3150.2.a.bj 1
20.e even 4 2 3150.2.g.c 2
21.c even 2 1 3920.2.a.t 1
24.f even 2 1 2240.2.a.n 1
24.h odd 2 1 2240.2.a.q 1
28.d even 2 1 4410.2.a.b 1
60.h even 2 1 350.2.a.b 1
60.l odd 4 2 350.2.c.b 2
84.h odd 2 1 490.2.a.h 1
84.j odd 6 2 490.2.e.c 2
84.n even 6 2 490.2.e.d 2
132.d odd 2 1 8470.2.a.j 1
420.o odd 2 1 2450.2.a.l 1
420.w even 4 2 2450.2.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 12.b even 2 1
350.2.a.b 1 60.h even 2 1
350.2.c.b 2 60.l odd 4 2
490.2.a.h 1 84.h odd 2 1
490.2.e.c 2 84.j odd 6 2
490.2.e.d 2 84.n even 6 2
560.2.a.d 1 3.b odd 2 1
630.2.a.d 1 4.b odd 2 1
2240.2.a.n 1 24.f even 2 1
2240.2.a.q 1 24.h odd 2 1
2450.2.a.l 1 420.o odd 2 1
2450.2.c.k 2 420.w even 4 2
2800.2.a.m 1 15.d odd 2 1
2800.2.g.n 2 15.e even 4 2
3150.2.a.bj 1 20.d odd 2 1
3150.2.g.c 2 20.e even 4 2
3920.2.a.t 1 21.c even 2 1
4410.2.a.b 1 28.d even 2 1
5040.2.a.bm 1 1.a even 1 1 trivial
8470.2.a.j 1 132.d odd 2 1

## Hecke kernels

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

 $$T_{11} - 4$$ $$T_{13} + 6$$ $$T_{17} + 2$$ $$T_{19}$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$2 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$8 + T$$
$37$ $$10 + T$$
$41$ $$2 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-2 + T$$
$59$ $$8 + T$$
$61$ $$14 + T$$
$67$ $$-12 + T$$
$71$ $$16 + T$$
$73$ $$-2 + T$$
$79$ $$-8 + T$$
$83$ $$-8 + T$$
$89$ $$10 + T$$
$97$ $$-2 + T$$