# Properties

 Label 504.2.p.c Level 504 Weight 2 Character orbit 504.p Analytic conductor 4.024 Analytic rank 0 Dimension 4 CM discriminant -168 Inner twists 8

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -2 q^{4} -\beta_{3} q^{7} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} -2 q^{4} -\beta_{3} q^{7} -2 \beta_{1} q^{8} -2 \beta_{3} q^{13} -\beta_{2} q^{14} + 4 q^{16} -2 \beta_{2} q^{17} + 2 \beta_{1} q^{23} -5 q^{25} -2 \beta_{2} q^{26} + 2 \beta_{3} q^{28} -4 \beta_{1} q^{29} -2 \beta_{3} q^{31} + 4 \beta_{1} q^{32} + 4 \beta_{3} q^{34} -2 \beta_{2} q^{41} + 2 q^{43} -4 q^{46} + 7 q^{49} -5 \beta_{1} q^{50} + 4 \beta_{3} q^{52} + 8 \beta_{1} q^{53} + 2 \beta_{2} q^{56} + 8 q^{58} + 4 \beta_{2} q^{59} -2 \beta_{3} q^{61} -2 \beta_{2} q^{62} -8 q^{64} -10 q^{67} + 4 \beta_{2} q^{68} -10 \beta_{1} q^{71} + 4 \beta_{3} q^{82} + 4 \beta_{2} q^{83} + 2 \beta_{1} q^{86} -2 \beta_{2} q^{89} + 14 q^{91} -4 \beta_{1} q^{92} + 7 \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 16q^{16} - 20q^{25} + 8q^{43} - 16q^{46} + 28q^{49} + 32q^{58} - 32q^{64} - 40q^{67} + 56q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## 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
 − 1.16372i 2.57794i 1.16372i − 2.57794i
1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
307.2 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 0 0
307.3 1.41421i 0 −2.00000 0 0 −2.64575 2.82843i 0 0
307.4 1.41421i 0 −2.00000 0 0 2.64575 2.82843i 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
168.e odd 2 1 CM by $$\Q(\sqrt{-42})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
8.d odd 2 1 inner
21.c even 2 1 inner
24.f even 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.c 4
3.b odd 2 1 inner 504.2.p.c 4
4.b odd 2 1 2016.2.p.c 4
7.b odd 2 1 inner 504.2.p.c 4
8.b even 2 1 2016.2.p.c 4
8.d odd 2 1 inner 504.2.p.c 4
12.b even 2 1 2016.2.p.c 4
21.c even 2 1 inner 504.2.p.c 4
24.f even 2 1 inner 504.2.p.c 4
24.h odd 2 1 2016.2.p.c 4
28.d even 2 1 2016.2.p.c 4
56.e even 2 1 inner 504.2.p.c 4
56.h odd 2 1 2016.2.p.c 4
84.h odd 2 1 2016.2.p.c 4
168.e odd 2 1 CM 504.2.p.c 4
168.i even 2 1 2016.2.p.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.c 4 1.a even 1 1 trivial
504.2.p.c 4 3.b odd 2 1 inner
504.2.p.c 4 7.b odd 2 1 inner
504.2.p.c 4 8.d odd 2 1 inner
504.2.p.c 4 21.c even 2 1 inner
504.2.p.c 4 24.f even 2 1 inner
504.2.p.c 4 56.e even 2 1 inner
504.2.p.c 4 168.e odd 2 1 CM
2016.2.p.c 4 4.b odd 2 1
2016.2.p.c 4 8.b even 2 1
2016.2.p.c 4 12.b even 2 1
2016.2.p.c 4 24.h odd 2 1
2016.2.p.c 4 28.d even 2 1
2016.2.p.c 4 56.h odd 2 1
2016.2.p.c 4 84.h odd 2 1
2016.2.p.c 4 168.i even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ 
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$( 1 + 11 T^{2} )^{4}$$
$13$ $$( 1 - 2 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 22 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 - 38 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 26 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 34 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 - 26 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 2 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 22 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 106 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 94 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 10 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 + 58 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 122 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 97 T^{2} )^{4}$$