# Properties

 Label 504.2.p.d Level 504 Weight 2 Character orbit 504.p Analytic conductor 4.024 Analytic rank 0 Dimension 4 CM discriminant -7 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: $$\Q(i, \sqrt{7})$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( 2 - \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} - 2 \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( 2 - \beta_{2} ) q^{4} + ( -1 + 2 \beta_{2} ) q^{7} + ( \beta_{1} - 2 \beta_{3} ) q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( 2 - 3 \beta_{2} ) q^{16} + ( 8 - 2 \beta_{2} ) q^{22} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{23} -5 q^{25} + ( 2 + 3 \beta_{2} ) q^{28} + ( -\beta_{1} + \beta_{3} ) q^{29} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{32} + ( -4 + 8 \beta_{2} ) q^{37} -12 q^{43} + ( 2 \beta_{1} - 8 \beta_{3} ) q^{44} -8 \beta_{2} q^{46} -7 q^{49} + 5 \beta_{3} q^{50} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{53} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{56} + 2 \beta_{2} q^{58} + ( -2 - 5 \beta_{2} ) q^{64} + 4 q^{67} + ( 8 \beta_{1} - 8 \beta_{3} ) q^{71} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{74} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{77} + ( -6 + 12 \beta_{2} ) q^{79} + 12 \beta_{3} q^{86} + ( 12 - 10 \beta_{2} ) q^{88} + 8 \beta_{1} q^{92} + 7 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{4} + O(q^{10})$$ $$4q + 6q^{4} + 2q^{16} + 28q^{22} - 20q^{25} + 14q^{28} - 48q^{43} - 16q^{46} - 28q^{49} + 4q^{58} - 18q^{64} + 16q^{67} + 28q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

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

## 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
 1.32288 − 0.500000i 1.32288 + 0.500000i −1.32288 − 0.500000i −1.32288 + 0.500000i
−1.32288 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i −1.32288 2.50000i 0 0
307.2 −1.32288 + 0.500000i 0 1.50000 1.32288i 0 0 2.64575i −1.32288 + 2.50000i 0 0
307.3 1.32288 0.500000i 0 1.50000 1.32288i 0 0 2.64575i 1.32288 2.50000i 0 0
307.4 1.32288 + 0.500000i 0 1.50000 + 1.32288i 0 0 2.64575i 1.32288 + 2.50000i 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 $$\Q(\sqrt{-7})$$
3.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
168.e odd 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.d 4
3.b odd 2 1 inner 504.2.p.d 4
4.b odd 2 1 2016.2.p.d 4
7.b odd 2 1 CM 504.2.p.d 4
8.b even 2 1 2016.2.p.d 4
8.d odd 2 1 inner 504.2.p.d 4
12.b even 2 1 2016.2.p.d 4
21.c even 2 1 inner 504.2.p.d 4
24.f even 2 1 inner 504.2.p.d 4
24.h odd 2 1 2016.2.p.d 4
28.d even 2 1 2016.2.p.d 4
56.e even 2 1 inner 504.2.p.d 4
56.h odd 2 1 2016.2.p.d 4
84.h odd 2 1 2016.2.p.d 4
168.e odd 2 1 inner 504.2.p.d 4
168.i even 2 1 2016.2.p.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.p.d 4 1.a even 1 1 trivial
504.2.p.d 4 3.b odd 2 1 inner
504.2.p.d 4 7.b odd 2 1 CM
504.2.p.d 4 8.d odd 2 1 inner
504.2.p.d 4 21.c even 2 1 inner
504.2.p.d 4 24.f even 2 1 inner
504.2.p.d 4 56.e even 2 1 inner
504.2.p.d 4 168.e odd 2 1 inner
2016.2.p.d 4 4.b odd 2 1
2016.2.p.d 4 8.b even 2 1
2016.2.p.d 4 12.b even 2 1
2016.2.p.d 4 24.h odd 2 1
2016.2.p.d 4 28.d even 2 1
2016.2.p.d 4 56.h odd 2 1
2016.2.p.d 4 84.h odd 2 1
2016.2.p.d 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 $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} - 28$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 4 T^{4}$$
$3$ 
$5$ $$( 1 + 5 T^{2} )^{4}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$( 1 - 6 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 - 17 T^{2} )^{4}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{4}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 + 12 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{4}$$
$61$ $$( 1 + 61 T^{2} )^{4}$$
$67$ $$( 1 - 4 T + 67 T^{2} )^{4}$$
$71$ $$( 1 + 114 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 - 89 T^{2} )^{4}$$
$97$ $$( 1 - 97 T^{2} )^{4}$$