# Properties

 Label 504.2.a.a Level $504$ Weight $2$ Character orbit 504.a Self dual yes Analytic conductor $4.024$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 504.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.02446026187$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} - q^{7}+O(q^{10})$$ q - 2 * q^5 - q^7 $$q - 2 q^{5} - q^{7} - 2 q^{11} + 2 q^{13} - 6 q^{17} - 4 q^{19} - 6 q^{23} - q^{25} - 4 q^{31} + 2 q^{35} + 10 q^{37} - 2 q^{41} - 4 q^{43} - 4 q^{47} + q^{49} + 12 q^{53} + 4 q^{55} - 12 q^{59} + 6 q^{61} - 4 q^{65} - 4 q^{67} + 14 q^{71} - 2 q^{73} + 2 q^{77} - 8 q^{79} + 16 q^{83} + 12 q^{85} + 6 q^{89} - 2 q^{91} + 8 q^{95} - 18 q^{97}+O(q^{100})$$ q - 2 * q^5 - q^7 - 2 * q^11 + 2 * q^13 - 6 * q^17 - 4 * q^19 - 6 * q^23 - q^25 - 4 * q^31 + 2 * q^35 + 10 * q^37 - 2 * q^41 - 4 * q^43 - 4 * q^47 + q^49 + 12 * q^53 + 4 * q^55 - 12 * q^59 + 6 * q^61 - 4 * q^65 - 4 * q^67 + 14 * q^71 - 2 * q^73 + 2 * q^77 - 8 * q^79 + 16 * q^83 + 12 * q^85 + 6 * q^89 - 2 * q^91 + 8 * q^95 - 18 * q^97

## 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 −2.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$

## 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 504.2.a.a 1
3.b odd 2 1 504.2.a.f yes 1
4.b odd 2 1 1008.2.a.f 1
7.b odd 2 1 3528.2.a.u 1
7.c even 3 2 3528.2.s.x 2
7.d odd 6 2 3528.2.s.i 2
8.b even 2 1 4032.2.a.bf 1
8.d odd 2 1 4032.2.a.bg 1
12.b even 2 1 1008.2.a.k 1
21.c even 2 1 3528.2.a.e 1
21.g even 6 2 3528.2.s.u 2
21.h odd 6 2 3528.2.s.f 2
24.f even 2 1 4032.2.a.l 1
24.h odd 2 1 4032.2.a.g 1
28.d even 2 1 7056.2.a.bt 1
84.h odd 2 1 7056.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.a.a 1 1.a even 1 1 trivial
504.2.a.f yes 1 3.b odd 2 1
1008.2.a.f 1 4.b odd 2 1
1008.2.a.k 1 12.b even 2 1
3528.2.a.e 1 21.c even 2 1
3528.2.a.u 1 7.b odd 2 1
3528.2.s.f 2 21.h odd 6 2
3528.2.s.i 2 7.d odd 6 2
3528.2.s.u 2 21.g even 6 2
3528.2.s.x 2 7.c even 3 2
4032.2.a.g 1 24.h odd 2 1
4032.2.a.l 1 24.f even 2 1
4032.2.a.bf 1 8.b even 2 1
4032.2.a.bg 1 8.d odd 2 1
7056.2.a.l 1 84.h odd 2 1
7056.2.a.bt 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{11} + 2$$ T11 + 2 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T + 1$$
$11$ $$T + 2$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T - 10$$
$41$ $$T + 2$$
$43$ $$T + 4$$
$47$ $$T + 4$$
$53$ $$T - 12$$
$59$ $$T + 12$$
$61$ $$T - 6$$
$67$ $$T + 4$$
$71$ $$T - 14$$
$73$ $$T + 2$$
$79$ $$T + 8$$
$83$ $$T - 16$$
$89$ $$T - 6$$
$97$ $$T + 18$$