# Properties

 Label 504.2.p.f Level 504 Weight 2 Character orbit 504.p Analytic conductor 4.024 Analytic rank 0 Dimension 4 CM no 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} -2 \beta_{1} q^{4} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} -2 \beta_{1} q^{4} + \beta_{3} q^{5} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -2 - 2 \beta_{1} ) q^{8} + ( -\beta_{2} + \beta_{3} ) q^{10} -2 q^{11} + \beta_{3} q^{13} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{14} -4 q^{16} + 2 \beta_{2} q^{17} -\beta_{2} q^{19} -2 \beta_{2} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} -4 \beta_{1} q^{23} + q^{25} + ( -\beta_{2} + \beta_{3} ) q^{26} + ( 2 - 2 \beta_{2} ) q^{28} + 4 \beta_{1} q^{29} -2 \beta_{3} q^{31} + ( -4 + 4 \beta_{1} ) q^{32} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( 6 + \beta_{2} ) q^{35} -8 \beta_{1} q^{37} + ( -\beta_{2} - \beta_{3} ) q^{38} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{40} -6 q^{43} + 4 \beta_{1} q^{44} + ( -4 - 4 \beta_{1} ) q^{46} -2 \beta_{3} q^{47} + ( 5 + 2 \beta_{2} ) q^{49} + ( 1 - \beta_{1} ) q^{50} -2 \beta_{2} q^{52} -4 \beta_{1} q^{53} -2 \beta_{3} q^{55} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{56} + ( 4 + 4 \beta_{1} ) q^{58} + \beta_{2} q^{59} + 3 \beta_{3} q^{61} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{62} + 8 \beta_{1} q^{64} + 6 q^{65} -2 q^{67} + 4 \beta_{3} q^{68} + ( 6 - 6 \beta_{1} + \beta_{2} + \beta_{3} ) q^{70} + 10 \beta_{1} q^{71} + 6 \beta_{2} q^{73} + ( -8 - 8 \beta_{1} ) q^{74} -2 \beta_{3} q^{76} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{77} + 6 \beta_{1} q^{79} -4 \beta_{3} q^{80} -\beta_{2} q^{83} + 12 \beta_{1} q^{85} + ( -6 + 6 \beta_{1} ) q^{86} + ( 4 + 4 \beta_{1} ) q^{88} + 6 \beta_{2} q^{89} + ( 6 + \beta_{2} ) q^{91} -8 q^{92} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{94} -6 \beta_{1} q^{95} -2 \beta_{2} q^{97} + ( 5 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 8q^{8} + O(q^{10})$$ $$4q + 4q^{2} - 8q^{8} - 8q^{11} + 4q^{14} - 16q^{16} - 8q^{22} + 4q^{25} + 8q^{28} - 16q^{32} + 24q^{35} - 24q^{43} - 16q^{46} + 20q^{49} + 4q^{50} + 8q^{56} + 16q^{58} + 24q^{65} - 8q^{67} + 24q^{70} - 32q^{74} - 24q^{86} + 16q^{88} + 24q^{91} - 32q^{92} + 20q^{98} + O(q^{100})$$

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

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

## 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
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000 1.00000i 0 2.00000i −2.44949 0 −2.44949 + 1.00000i −2.00000 2.00000i 0 −2.44949 + 2.44949i
307.2 1.00000 1.00000i 0 2.00000i 2.44949 0 2.44949 + 1.00000i −2.00000 2.00000i 0 2.44949 2.44949i
307.3 1.00000 + 1.00000i 0 2.00000i −2.44949 0 −2.44949 1.00000i −2.00000 + 2.00000i 0 −2.44949 2.44949i
307.4 1.00000 + 1.00000i 0 2.00000i 2.44949 0 2.44949 1.00000i −2.00000 + 2.00000i 0 2.44949 + 2.44949i
 $$n$$: 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 inner
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.f 4
3.b odd 2 1 56.2.e.b 4
4.b odd 2 1 2016.2.p.e 4
7.b odd 2 1 inner 504.2.p.f 4
8.b even 2 1 2016.2.p.e 4
8.d odd 2 1 inner 504.2.p.f 4
12.b even 2 1 224.2.e.b 4
21.c even 2 1 56.2.e.b 4
21.g even 6 2 392.2.m.f 8
21.h odd 6 2 392.2.m.f 8
24.f even 2 1 56.2.e.b 4
24.h odd 2 1 224.2.e.b 4
28.d even 2 1 2016.2.p.e 4
48.i odd 4 1 1792.2.f.e 4
48.i odd 4 1 1792.2.f.f 4
48.k even 4 1 1792.2.f.e 4
48.k even 4 1 1792.2.f.f 4
56.e even 2 1 inner 504.2.p.f 4
56.h odd 2 1 2016.2.p.e 4
84.h odd 2 1 224.2.e.b 4
84.j odd 6 2 1568.2.q.e 8
84.n even 6 2 1568.2.q.e 8
168.e odd 2 1 56.2.e.b 4
168.i even 2 1 224.2.e.b 4
168.s odd 6 2 1568.2.q.e 8
168.v even 6 2 392.2.m.f 8
168.ba even 6 2 1568.2.q.e 8
168.be odd 6 2 392.2.m.f 8
336.v odd 4 1 1792.2.f.e 4
336.v odd 4 1 1792.2.f.f 4
336.y even 4 1 1792.2.f.e 4
336.y even 4 1 1792.2.f.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.e.b 4 3.b odd 2 1
56.2.e.b 4 21.c even 2 1
56.2.e.b 4 24.f even 2 1
56.2.e.b 4 168.e odd 2 1
224.2.e.b 4 12.b even 2 1
224.2.e.b 4 24.h odd 2 1
224.2.e.b 4 84.h odd 2 1
224.2.e.b 4 168.i even 2 1
392.2.m.f 8 21.g even 6 2
392.2.m.f 8 21.h odd 6 2
392.2.m.f 8 168.v even 6 2
392.2.m.f 8 168.be odd 6 2
504.2.p.f 4 1.a even 1 1 trivial
504.2.p.f 4 7.b odd 2 1 inner
504.2.p.f 4 8.d odd 2 1 inner
504.2.p.f 4 56.e even 2 1 inner
1568.2.q.e 8 84.j odd 6 2
1568.2.q.e 8 84.n even 6 2
1568.2.q.e 8 168.s odd 6 2
1568.2.q.e 8 168.ba even 6 2
1792.2.f.e 4 48.i odd 4 1
1792.2.f.e 4 48.k even 4 1
1792.2.f.e 4 336.v odd 4 1
1792.2.f.e 4 336.y even 4 1
1792.2.f.f 4 48.i odd 4 1
1792.2.f.f 4 48.k even 4 1
1792.2.f.f 4 336.v odd 4 1
1792.2.f.f 4 336.y even 4 1
2016.2.p.e 4 4.b odd 2 1
2016.2.p.e 4 8.b even 2 1
2016.2.p.e 4 28.d even 2 1
2016.2.p.e 4 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 $$S_{2}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{2} - 6$$ $$T_{11} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 2 T^{2} )^{2}$$
$3$ 
$5$ $$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 2 T + 11 T^{2} )^{4}$$
$13$ $$( 1 + 20 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 41 T^{2} )^{4}$$
$43$ $$( 1 + 6 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 112 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 68 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 2 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 70 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 38 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$