# Properties

 Label 504.2.a.f Level 504 Weight 2 Character orbit 504.a Self dual yes Analytic conductor 4.024 Analytic rank 0 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: $$0$$ 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

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

## 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 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

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

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

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

 $$T_{5} - 2$$ $$T_{11} - 2$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

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