# Properties

 Label 504.2.c.d Level 504 Weight 2 Character orbit 504.c Analytic conductor 4.024 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2312.1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} - q^{7} + ( 2 + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} - q^{7} + ( 2 + \beta_{3} ) q^{8} + ( -2 + \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{2} - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{13} -\beta_{1} q^{14} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{16} -2 q^{17} + ( \beta_{1} - \beta_{2} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{20} -2 \beta_{3} q^{22} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -2 + \beta_{2} + \beta_{3} ) q^{26} -\beta_{2} q^{28} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{32} -2 \beta_{1} q^{34} + ( -\beta_{1} - \beta_{3} ) q^{35} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -2 + \beta_{2} - \beta_{3} ) q^{38} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{40} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( \beta_{2} + \beta_{3} ) q^{43} + ( 4 - 2 \beta_{3} ) q^{44} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{46} + q^{49} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{52} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( -2 - \beta_{3} ) q^{56} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{58} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{59} + ( \beta_{1} + \beta_{3} ) q^{61} + ( 8 - 4 \beta_{1} + 4 \beta_{2} ) q^{62} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{64} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( -6 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{67} -2 \beta_{2} q^{68} + ( 2 - \beta_{2} - \beta_{3} ) q^{70} + 8 q^{71} -6 q^{73} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{74} + ( 4 - 2 \beta_{1} ) q^{76} + ( \beta_{2} + \beta_{3} ) q^{77} + ( -8 - 4 \beta_{1} ) q^{80} + ( 8 + 2 \beta_{1} + 4 \beta_{2} ) q^{82} + ( \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{85} + 2 \beta_{3} q^{86} + ( 4 + 4 \beta_{1} - 2 \beta_{3} ) q^{88} + ( 10 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{91} + ( 4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{92} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{95} + ( 6 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} + q^{4} - 4q^{7} + 7q^{8} + O(q^{10})$$ $$4q + q^{2} + q^{4} - 4q^{7} + 7q^{8} - 8q^{10} - q^{14} - 7q^{16} - 8q^{17} - 4q^{20} + 2q^{22} - 4q^{23} - 8q^{25} - 8q^{26} - q^{28} - 8q^{31} - 9q^{32} - 2q^{34} - 6q^{38} - 20q^{40} + 16q^{41} + 18q^{44} + 16q^{46} + 4q^{49} - 19q^{50} - 4q^{52} + 24q^{55} - 7q^{56} - 12q^{58} + 32q^{62} + q^{64} - 28q^{65} - 2q^{68} + 8q^{70} + 32q^{71} - 24q^{73} - 12q^{74} + 14q^{76} - 36q^{80} + 38q^{82} - 2q^{86} + 22q^{88} + 32q^{89} + 16q^{92} - 4q^{95} + 16q^{97} + q^{98} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
253.1
 −0.780776 − 1.17915i −0.780776 + 1.17915i 1.28078 − 0.599676i 1.28078 + 0.599676i
−0.780776 1.17915i 0 −0.780776 + 1.84130i 1.69614i 0 −1.00000 2.78078 0.516994i 0 −2.00000 + 1.32431i
253.2 −0.780776 + 1.17915i 0 −0.780776 1.84130i 1.69614i 0 −1.00000 2.78078 + 0.516994i 0 −2.00000 1.32431i
253.3 1.28078 0.599676i 0 1.28078 1.53610i 3.33513i 0 −1.00000 0.719224 2.73546i 0 −2.00000 4.27156i
253.4 1.28078 + 0.599676i 0 1.28078 + 1.53610i 3.33513i 0 −1.00000 0.719224 + 2.73546i 0 −2.00000 + 4.27156i
 $$n$$: 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
8.b 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.c.d 4
3.b odd 2 1 56.2.b.b 4
4.b odd 2 1 2016.2.c.c 4
8.b even 2 1 inner 504.2.c.d 4
8.d odd 2 1 2016.2.c.c 4
12.b even 2 1 224.2.b.b 4
21.c even 2 1 392.2.b.c 4
21.g even 6 2 392.2.p.e 8
21.h odd 6 2 392.2.p.f 8
24.f even 2 1 224.2.b.b 4
24.h odd 2 1 56.2.b.b 4
48.i odd 4 2 1792.2.a.x 4
48.k even 4 2 1792.2.a.v 4
84.h odd 2 1 1568.2.b.d 4
84.j odd 6 2 1568.2.t.e 8
84.n even 6 2 1568.2.t.d 8
168.e odd 2 1 1568.2.b.d 4
168.i even 2 1 392.2.b.c 4
168.s odd 6 2 392.2.p.f 8
168.v even 6 2 1568.2.t.d 8
168.ba even 6 2 392.2.p.e 8
168.be odd 6 2 1568.2.t.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 3.b odd 2 1
56.2.b.b 4 24.h odd 2 1
224.2.b.b 4 12.b even 2 1
224.2.b.b 4 24.f even 2 1
392.2.b.c 4 21.c even 2 1
392.2.b.c 4 168.i even 2 1
392.2.p.e 8 21.g even 6 2
392.2.p.e 8 168.ba even 6 2
392.2.p.f 8 21.h odd 6 2
392.2.p.f 8 168.s odd 6 2
504.2.c.d 4 1.a even 1 1 trivial
504.2.c.d 4 8.b even 2 1 inner
1568.2.b.d 4 84.h odd 2 1
1568.2.b.d 4 168.e odd 2 1
1568.2.t.d 8 84.n even 6 2
1568.2.t.d 8 168.v even 6 2
1568.2.t.e 8 84.j odd 6 2
1568.2.t.e 8 168.be odd 6 2
1792.2.a.v 4 48.k even 4 2
1792.2.a.x 4 48.i odd 4 2
2016.2.c.c 4 4.b odd 2 1
2016.2.c.c 4 8.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{2} + 32$$ $$T_{11}^{4} + 20 T_{11}^{2} + 32$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T - 2 T^{3} + 4 T^{4}$$
$3$ 1
$5$ $$1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8}$$
$13$ $$1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8}$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{4}$$
$19$ $$1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8}$$
$23$ $$( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} )^{2}$$
$29$ $$1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8}$$
$31$ $$( 1 + 4 T - 2 T^{2} + 124 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 - 8 T + 30 T^{2} - 328 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8}$$
$67$ $$1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 8 T + 71 T^{2} )^{4}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{4}$$
$79$ $$( 1 + 79 T^{2} )^{4}$$
$83$ $$1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 - 16 T + 174 T^{2} - 1424 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2}$$