# Properties

 Label 504.2.p.b 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: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{12}^{3} ) q^{2} -2 \zeta_{12}^{3} q^{4} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + 2 q^{11} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{14} -4 q^{16} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} + ( 4 - 8 \zeta_{12}^{2} ) q^{20} + ( -2 + 2 \zeta_{12}^{3} ) q^{22} -2 \zeta_{12}^{3} q^{23} + 7 q^{25} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( -2 - 4 \zeta_{12}^{2} ) q^{28} + 8 \zeta_{12}^{3} q^{29} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12}^{3} ) q^{32} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + ( 10 - 8 \zeta_{12}^{2} ) q^{35} + 4 \zeta_{12}^{3} q^{37} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{38} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{40} + ( -6 + 12 \zeta_{12}^{2} ) q^{41} + 6 q^{43} -4 \zeta_{12}^{3} q^{44} + ( 2 + 2 \zeta_{12}^{3} ) q^{46} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -7 + 7 \zeta_{12}^{3} ) q^{50} + ( -4 + 8 \zeta_{12}^{2} ) q^{52} + 4 \zeta_{12}^{3} q^{53} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{55} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} + ( -8 - 8 \zeta_{12}^{3} ) q^{58} + ( 4 - 8 \zeta_{12}^{2} ) q^{59} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{61} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} -12 q^{65} + 10 q^{67} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{68} + ( -10 + 8 \zeta_{12} + 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{70} + 2 \zeta_{12}^{3} q^{71} + ( -4 - 4 \zeta_{12}^{3} ) q^{74} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{76} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + 12 \zeta_{12}^{3} q^{79} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{80} + ( 6 - 12 \zeta_{12} - 12 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{82} + ( -4 + 8 \zeta_{12}^{2} ) q^{83} -12 \zeta_{12}^{3} q^{85} + ( -6 + 6 \zeta_{12}^{3} ) q^{86} + ( 4 + 4 \zeta_{12}^{3} ) q^{88} + ( -6 + 12 \zeta_{12}^{2} ) q^{89} + ( -10 + 8 \zeta_{12}^{2} ) q^{91} -4 q^{92} + ( 4 + 8 \zeta_{12} - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{94} -24 \zeta_{12}^{3} q^{95} + ( 8 - 16 \zeta_{12}^{2} ) q^{97} + ( -3 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 8q^{8} + O(q^{10})$$ $$4q - 4q^{2} + 8q^{8} + 8q^{11} + 8q^{14} - 16q^{16} - 8q^{22} + 28q^{25} - 16q^{28} + 16q^{32} + 24q^{35} + 24q^{43} + 8q^{46} - 4q^{49} - 28q^{50} + 16q^{56} - 32q^{58} - 48q^{65} + 40q^{67} - 24q^{70} - 16q^{74} - 24q^{86} + 16q^{88} - 24q^{91} - 16q^{92} + 4q^{98} + O(q^{100})$$

## 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
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
−1.00000 1.00000i 0 2.00000i −3.46410 0 −1.73205 + 2.00000i 2.00000 2.00000i 0 3.46410 + 3.46410i
307.2 −1.00000 1.00000i 0 2.00000i 3.46410 0 1.73205 + 2.00000i 2.00000 2.00000i 0 −3.46410 3.46410i
307.3 −1.00000 + 1.00000i 0 2.00000i −3.46410 0 −1.73205 2.00000i 2.00000 + 2.00000i 0 3.46410 3.46410i
307.4 −1.00000 + 1.00000i 0 2.00000i 3.46410 0 1.73205 2.00000i 2.00000 + 2.00000i 0 −3.46410 + 3.46410i
 $$n$$: 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.b 4
3.b odd 2 1 504.2.p.e yes 4
4.b odd 2 1 2016.2.p.b 4
7.b odd 2 1 inner 504.2.p.b 4
8.b even 2 1 2016.2.p.b 4
8.d odd 2 1 inner 504.2.p.b 4
12.b even 2 1 2016.2.p.f 4
21.c even 2 1 504.2.p.e yes 4
24.f even 2 1 504.2.p.e yes 4
24.h odd 2 1 2016.2.p.f 4
28.d even 2 1 2016.2.p.b 4
56.e even 2 1 inner 504.2.p.b 4
56.h odd 2 1 2016.2.p.b 4
84.h odd 2 1 2016.2.p.f 4
168.e odd 2 1 504.2.p.e yes 4
168.i even 2 1 2016.2.p.f 4

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

 $$T_{5}^{2} - 12$$ $$T_{11} - 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} )^{2}$$
$3$ 1
$5$ $$( 1 - 2 T^{2} + 25 T^{4} )^{2}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 2 T + 11 T^{2} )^{4}$$
$13$ $$( 1 + 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 22 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 10 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 42 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 6 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 50 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 58 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 26 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 6 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 46 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 70 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 14 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 10 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 138 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 73 T^{2} )^{4}$$
$79$ $$( 1 - 14 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 118 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 70 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2}$$