# Properties

 Label 504.2.a.b 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: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.b 1
3.b odd 2 1 168.2.a.b 1
4.b odd 2 1 1008.2.a.e 1
7.b odd 2 1 3528.2.a.w 1
7.c even 3 2 3528.2.s.v 2
7.d odd 6 2 3528.2.s.h 2
8.b even 2 1 4032.2.a.be 1
8.d odd 2 1 4032.2.a.bj 1
12.b even 2 1 336.2.a.c 1
15.d odd 2 1 4200.2.a.i 1
15.e even 4 2 4200.2.t.m 2
21.c even 2 1 1176.2.a.a 1
21.g even 6 2 1176.2.q.j 2
21.h odd 6 2 1176.2.q.b 2
24.f even 2 1 1344.2.a.n 1
24.h odd 2 1 1344.2.a.c 1
28.d even 2 1 7056.2.a.br 1
48.i odd 4 2 5376.2.c.bd 2
48.k even 4 2 5376.2.c.f 2
60.h even 2 1 8400.2.a.bx 1
84.h odd 2 1 2352.2.a.q 1
84.j odd 6 2 2352.2.q.j 2
84.n even 6 2 2352.2.q.o 2
168.e odd 2 1 9408.2.a.bc 1
168.i even 2 1 9408.2.a.cy 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.b 1 3.b odd 2 1
336.2.a.c 1 12.b even 2 1
504.2.a.b 1 1.a even 1 1 trivial
1008.2.a.e 1 4.b odd 2 1
1176.2.a.a 1 21.c even 2 1
1176.2.q.b 2 21.h odd 6 2
1176.2.q.j 2 21.g even 6 2
1344.2.a.c 1 24.h odd 2 1
1344.2.a.n 1 24.f even 2 1
2352.2.a.q 1 84.h odd 2 1
2352.2.q.j 2 84.j odd 6 2
2352.2.q.o 2 84.n even 6 2
3528.2.a.w 1 7.b odd 2 1
3528.2.s.h 2 7.d odd 6 2
3528.2.s.v 2 7.c even 3 2
4032.2.a.be 1 8.b even 2 1
4032.2.a.bj 1 8.d odd 2 1
4200.2.a.i 1 15.d odd 2 1
4200.2.t.m 2 15.e even 4 2
5376.2.c.f 2 48.k even 4 2
5376.2.c.bd 2 48.i odd 4 2
7056.2.a.br 1 28.d even 2 1
8400.2.a.bx 1 60.h even 2 1
9408.2.a.bc 1 168.e odd 2 1
9408.2.a.cy 1 168.i even 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(504))$$:

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

## Hecke Characteristic Polynomials

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