# Properties

 Label 504.2.cj.a Level 504 Weight 2 Character orbit 504.cj Analytic conductor 4.024 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.cj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12960000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} + \beta_{5} ) q^{2} + \beta_{6} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 + 3 \beta_{2} ) q^{7} + ( \beta_{4} - 2 \beta_{7} ) q^{8} +O(q^{10})$$ $$q + ( \beta_{4} + \beta_{5} ) q^{2} + \beta_{6} q^{4} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{5} + ( 1 + 3 \beta_{2} ) q^{7} + ( \beta_{4} - 2 \beta_{7} ) q^{8} + ( 2 + 2 \beta_{2} + \beta_{6} ) q^{10} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{11} + ( \beta_{4} - 2 \beta_{5} ) q^{14} + ( -\beta_{1} - 4 \beta_{2} ) q^{16} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{19} + ( 3 \beta_{4} - 2 \beta_{7} ) q^{20} + ( -2 - \beta_{1} - \beta_{6} ) q^{22} + ( -4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{23} + ( -3 \beta_{1} - 2 \beta_{6} ) q^{28} + ( -\beta_{4} - \beta_{7} ) q^{29} + ( 1 + \beta_{2} ) q^{31} + ( 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{32} + ( 4 - 2 \beta_{1} - 2 \beta_{6} ) q^{34} + ( -2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{7} ) q^{35} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -8 \beta_{3} + 2 \beta_{5} + 8 \beta_{7} ) q^{38} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{40} + ( 6 \beta_{4} - 6 \beta_{7} ) q^{41} + ( 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{44} + ( -8 - 8 \beta_{2} + 4 \beta_{6} ) q^{46} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -8 - 3 \beta_{2} ) q^{49} + ( 5 \beta_{3} + 5 \beta_{5} - 5 \beta_{7} ) q^{53} -5 q^{55} + ( 6 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} ) q^{56} + ( \beta_{1} - 2 \beta_{2} ) q^{58} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{59} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{61} + \beta_{4} q^{62} + ( 4 + 3 \beta_{1} + 3 \beta_{6} ) q^{64} + ( 2 + 2 \beta_{2} - 4 \beta_{6} ) q^{67} + ( 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{68} + ( -4 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} ) q^{70} + ( -6 \beta_{4} + 6 \beta_{7} ) q^{71} + ( 10 + 10 \beta_{2} ) q^{73} + ( -8 \beta_{3} + 2 \beta_{5} + 8 \beta_{7} ) q^{74} + ( -16 + 2 \beta_{1} + 2 \beta_{6} ) q^{76} + ( -\beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{77} -13 \beta_{2} q^{79} + ( 6 \beta_{3} + \beta_{5} - 6 \beta_{7} ) q^{80} + ( -6 \beta_{1} - 12 \beta_{2} ) q^{82} + ( 5 \beta_{4} + 5 \beta_{7} ) q^{83} + ( 2 - 4 \beta_{1} - 4 \beta_{6} ) q^{85} + ( 4 + 4 \beta_{2} - 3 \beta_{6} ) q^{88} + ( 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{89} + ( -4 \beta_{4} - 8 \beta_{7} ) q^{92} + ( 4 + 4 \beta_{2} - 2 \beta_{6} ) q^{94} + ( -10 \beta_{3} + 10 \beta_{5} + 10 \beta_{7} ) q^{95} - q^{97} + ( -8 \beta_{4} - 5 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{4} - 4q^{7} + O(q^{10})$$ $$8q + 2q^{4} - 4q^{7} + 10q^{10} + 14q^{16} - 20q^{22} - 10q^{28} + 4q^{31} + 24q^{34} + 10q^{40} - 24q^{46} - 52q^{49} - 40q^{55} + 10q^{58} + 44q^{64} - 50q^{70} + 40q^{73} - 120q^{76} + 52q^{79} + 36q^{82} + 10q^{88} + 12q^{94} - 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} - 16 \nu^{2} + 19$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{6} + 8 \nu^{4} - 24 \nu^{2} + 1$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 12 \nu^{3} + 11 \nu$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} - 6 \nu^{3} - 3 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 16 \nu^{5} - 40 \nu^{3} + 23 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{6} + 16 \nu^{4} - 40 \nu^{2} - 3$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$-\nu^{7} + 3 \nu^{5} - 8 \nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 5 \beta_{2} + \beta_{1} - 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 4 \beta_{5} - \beta_{4} - 4 \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - 3 \beta_{2} + 2 \beta_{1} - 4$$ $$\nu^{5}$$ $$=$$ $$($$$$11 \beta_{7} - 2 \beta_{5} - 13 \beta_{4} - 13 \beta_{3}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-8 \beta_{6} + 8 \beta_{2} + 8 \beta_{1} - 23$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$34 \beta_{7} - 34 \beta_{5} - 29 \beta_{4} - 5 \beta_{3}$$$$)/3$$

## Character values

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

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −1.40126 + 0.809017i −0.535233 + 0.309017i 0.535233 − 0.309017i 1.40126 − 0.809017i −1.40126 − 0.809017i −0.535233 − 0.309017i 0.535233 + 0.309017i 1.40126 + 0.809017i
−1.40126 + 0.190983i 0 1.92705 0.535233i −1.93649 1.11803i 0 −0.500000 + 2.59808i −2.59808 + 1.11803i 0 2.92705 + 1.19682i
37.2 −0.535233 1.30902i 0 −1.42705 + 1.40126i −1.93649 1.11803i 0 −0.500000 + 2.59808i 2.59808 + 1.11803i 0 −0.427051 + 3.13331i
37.3 0.535233 + 1.30902i 0 −1.42705 + 1.40126i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i −2.59808 1.11803i 0 −0.427051 + 3.13331i
37.4 1.40126 0.190983i 0 1.92705 0.535233i 1.93649 + 1.11803i 0 −0.500000 + 2.59808i 2.59808 1.11803i 0 2.92705 + 1.19682i
109.1 −1.40126 0.190983i 0 1.92705 + 0.535233i −1.93649 + 1.11803i 0 −0.500000 2.59808i −2.59808 1.11803i 0 2.92705 1.19682i
109.2 −0.535233 + 1.30902i 0 −1.42705 1.40126i −1.93649 + 1.11803i 0 −0.500000 2.59808i 2.59808 1.11803i 0 −0.427051 3.13331i
109.3 0.535233 1.30902i 0 −1.42705 1.40126i 1.93649 1.11803i 0 −0.500000 2.59808i −2.59808 + 1.11803i 0 −0.427051 3.13331i
109.4 1.40126 + 0.190983i 0 1.92705 + 0.535233i 1.93649 1.11803i 0 −0.500000 2.59808i 2.59808 + 1.11803i 0 2.92705 1.19682i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.p even 6 1 inner
168.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cj.a 8
3.b odd 2 1 inner 504.2.cj.a 8
4.b odd 2 1 2016.2.cr.a 8
7.c even 3 1 inner 504.2.cj.a 8
8.b even 2 1 inner 504.2.cj.a 8
8.d odd 2 1 2016.2.cr.a 8
12.b even 2 1 2016.2.cr.a 8
21.h odd 6 1 inner 504.2.cj.a 8
24.f even 2 1 2016.2.cr.a 8
24.h odd 2 1 inner 504.2.cj.a 8
28.g odd 6 1 2016.2.cr.a 8
56.k odd 6 1 2016.2.cr.a 8
56.p even 6 1 inner 504.2.cj.a 8
84.n even 6 1 2016.2.cr.a 8
168.s odd 6 1 inner 504.2.cj.a 8
168.v even 6 1 2016.2.cr.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cj.a 8 1.a even 1 1 trivial
504.2.cj.a 8 3.b odd 2 1 inner
504.2.cj.a 8 7.c even 3 1 inner
504.2.cj.a 8 8.b even 2 1 inner
504.2.cj.a 8 21.h odd 6 1 inner
504.2.cj.a 8 24.h odd 2 1 inner
504.2.cj.a 8 56.p even 6 1 inner
504.2.cj.a 8 168.s odd 6 1 inner
2016.2.cr.a 8 4.b odd 2 1
2016.2.cr.a 8 8.d odd 2 1
2016.2.cr.a 8 12.b even 2 1
2016.2.cr.a 8 24.f even 2 1
2016.2.cr.a 8 28.g odd 6 1
2016.2.cr.a 8 56.k odd 6 1
2016.2.cr.a 8 84.n even 6 1
2016.2.cr.a 8 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 5 T_{5}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} - 3 T^{4} - 4 T^{6} + 16 T^{8}$$
$3$ 
$5$ $$( 1 + 5 T^{2} )^{4}( 1 - 5 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 + T + 7 T^{2} )^{4}$$
$11$ $$( 1 + 17 T^{2} + 168 T^{4} + 2057 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 13 T^{2} )^{8}$$
$17$ $$( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} )^{2}( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 2 T^{2} - 525 T^{4} + 1058 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 53 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{4}$$
$37$ $$( 1 + 14 T^{2} - 1173 T^{4} + 19166 T^{6} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 26 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 43 T^{2} )^{8}$$
$47$ $$( 1 - 82 T^{2} + 4515 T^{4} - 181138 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 19 T^{2} - 2448 T^{4} - 53371 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 113 T^{2} + 9288 T^{4} + 393353 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 62 T^{2} + 123 T^{4} + 230702 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 74 T^{2} + 987 T^{4} + 332186 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 34 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 17 T + 73 T^{2} )^{4}( 1 + 7 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 41 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 + 14 T^{2} - 7725 T^{4} + 110894 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + T + 97 T^{2} )^{8}$$