# Properties

 Label 504.2.p.a Level 504 Weight 2 Character orbit 504.p Analytic conductor 4.024 Analytic rank 0 Dimension 2 CM discriminant -7 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-7})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( 3 - \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( -1 - \beta ) q^{4} + ( -1 + 2 \beta ) q^{7} + ( 3 - \beta ) q^{8} + 4 q^{11} + ( -3 - \beta ) q^{14} + ( -1 + 3 \beta ) q^{16} + ( -4 + 4 \beta ) q^{22} + ( -2 + 4 \beta ) q^{23} -5 q^{25} + ( 5 - 3 \beta ) q^{28} + ( -4 + 8 \beta ) q^{29} + ( -5 - \beta ) q^{32} + ( -4 + 8 \beta ) q^{37} + 12 q^{43} + ( -4 - 4 \beta ) q^{44} + ( -6 - 2 \beta ) q^{46} -7 q^{49} + ( 5 - 5 \beta ) q^{50} + ( 4 - 8 \beta ) q^{53} + ( 1 + 5 \beta ) q^{56} + ( -12 - 4 \beta ) q^{58} + ( 7 - 5 \beta ) q^{64} + 4 q^{67} + ( 2 - 4 \beta ) q^{71} + ( -12 - 4 \beta ) q^{74} + ( -4 + 8 \beta ) q^{77} + ( 6 - 12 \beta ) q^{79} + ( -12 + 12 \beta ) q^{86} + ( 12 - 4 \beta ) q^{88} + ( 10 - 6 \beta ) q^{92} + ( 7 - 7 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 3q^{4} + 5q^{8} + O(q^{10})$$ $$2q - q^{2} - 3q^{4} + 5q^{8} + 8q^{11} - 7q^{14} + q^{16} - 4q^{22} - 10q^{25} + 7q^{28} - 11q^{32} + 24q^{43} - 12q^{44} - 14q^{46} - 14q^{49} + 5q^{50} + 7q^{56} - 28q^{58} + 9q^{64} + 8q^{67} - 28q^{74} - 12q^{86} + 20q^{88} + 14q^{92} + 7q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 0.5 − 1.32288i 0.5 + 1.32288i
−0.500000 1.32288i 0 −1.50000 + 1.32288i 0 0 2.64575i 2.50000 + 1.32288i 0 0
307.2 −0.500000 + 1.32288i 0 −1.50000 1.32288i 0 0 2.64575i 2.50000 1.32288i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
8.d odd 2 1 inner
56.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.p.a 2
3.b odd 2 1 56.2.e.a 2
4.b odd 2 1 2016.2.p.a 2
7.b odd 2 1 CM 504.2.p.a 2
8.b even 2 1 2016.2.p.a 2
8.d odd 2 1 inner 504.2.p.a 2
12.b even 2 1 224.2.e.a 2
21.c even 2 1 56.2.e.a 2
21.g even 6 2 392.2.m.a 4
21.h odd 6 2 392.2.m.a 4
24.f even 2 1 56.2.e.a 2
24.h odd 2 1 224.2.e.a 2
28.d even 2 1 2016.2.p.a 2
48.i odd 4 2 1792.2.f.d 4
48.k even 4 2 1792.2.f.d 4
56.e even 2 1 inner 504.2.p.a 2
56.h odd 2 1 2016.2.p.a 2
84.h odd 2 1 224.2.e.a 2
84.j odd 6 2 1568.2.q.a 4
84.n even 6 2 1568.2.q.a 4
168.e odd 2 1 56.2.e.a 2
168.i even 2 1 224.2.e.a 2
168.s odd 6 2 1568.2.q.a 4
168.v even 6 2 392.2.m.a 4
168.ba even 6 2 1568.2.q.a 4
168.be odd 6 2 392.2.m.a 4
336.v odd 4 2 1792.2.f.d 4
336.y even 4 2 1792.2.f.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.a 2 3.b odd 2 1
56.2.e.a 2 21.c even 2 1
56.2.e.a 2 24.f even 2 1
56.2.e.a 2 168.e odd 2 1
224.2.e.a 2 12.b even 2 1
224.2.e.a 2 24.h odd 2 1
224.2.e.a 2 84.h odd 2 1
224.2.e.a 2 168.i even 2 1
392.2.m.a 4 21.g even 6 2
392.2.m.a 4 21.h odd 6 2
392.2.m.a 4 168.v even 6 2
392.2.m.a 4 168.be odd 6 2
504.2.p.a 2 1.a even 1 1 trivial
504.2.p.a 2 7.b odd 2 1 CM
504.2.p.a 2 8.d odd 2 1 inner
504.2.p.a 2 56.e even 2 1 inner
1568.2.q.a 4 84.j odd 6 2
1568.2.q.a 4 84.n even 6 2
1568.2.q.a 4 168.s odd 6 2
1568.2.q.a 4 168.ba even 6 2
1792.2.f.d 4 48.i odd 4 2
1792.2.f.d 4 48.k even 4 2
1792.2.f.d 4 336.v odd 4 2
1792.2.f.d 4 336.y even 4 2
2016.2.p.a 2 4.b odd 2 1
2016.2.p.a 2 8.b even 2 1
2016.2.p.a 2 28.d even 2 1
2016.2.p.a 2 56.h odd 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11} - 4$$

## Hecke Characteristic Polynomials

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