# Properties

 Label 504.2.c.a Level 504 Weight 2 Character orbit 504.c Analytic conductor 4.024 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -2 q^{4} -\beta q^{5} + q^{7} -2 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -2 q^{4} -\beta q^{5} + q^{7} -2 \beta q^{8} + 2 q^{10} + 2 \beta q^{11} + 3 \beta q^{13} + \beta q^{14} + 4 q^{16} + 6 q^{17} + 3 \beta q^{19} + 2 \beta q^{20} -4 q^{22} + 6 q^{23} + 3 q^{25} -6 q^{26} -2 q^{28} + 2 \beta q^{29} -4 q^{31} + 4 \beta q^{32} + 6 \beta q^{34} -\beta q^{35} -6 \beta q^{37} -6 q^{38} -4 q^{40} -6 q^{41} + 6 \beta q^{43} -4 \beta q^{44} + 6 \beta q^{46} + q^{49} + 3 \beta q^{50} -6 \beta q^{52} -4 \beta q^{53} + 4 q^{55} -2 \beta q^{56} -4 q^{58} -\beta q^{59} -9 \beta q^{61} -4 \beta q^{62} -8 q^{64} + 6 q^{65} -12 q^{68} + 2 q^{70} + 2 q^{73} + 12 q^{74} -6 \beta q^{76} + 2 \beta q^{77} + 8 q^{79} -4 \beta q^{80} -6 \beta q^{82} + 11 \beta q^{83} -6 \beta q^{85} -12 q^{86} + 8 q^{88} -6 q^{89} + 3 \beta q^{91} -12 q^{92} + 6 q^{95} -10 q^{97} + \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 2q^{7} + O(q^{10})$$ $$2q - 4q^{4} + 2q^{7} + 4q^{10} + 8q^{16} + 12q^{17} - 8q^{22} + 12q^{23} + 6q^{25} - 12q^{26} - 4q^{28} - 8q^{31} - 12q^{38} - 8q^{40} - 12q^{41} + 2q^{49} + 8q^{55} - 8q^{58} - 16q^{64} + 12q^{65} - 24q^{68} + 4q^{70} + 4q^{73} + 24q^{74} + 16q^{79} - 24q^{86} + 16q^{88} - 12q^{89} - 24q^{92} + 12q^{95} - 20q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
253.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 1.41421i 0 1.00000 2.82843i 0 2.00000
253.2 1.41421i 0 −2.00000 1.41421i 0 1.00000 2.82843i 0 2.00000
 $$n$$: 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.a 2
3.b odd 2 1 56.2.b.a 2
4.b odd 2 1 2016.2.c.a 2
8.b even 2 1 inner 504.2.c.a 2
8.d odd 2 1 2016.2.c.a 2
12.b even 2 1 224.2.b.a 2
21.c even 2 1 392.2.b.b 2
21.g even 6 2 392.2.p.b 4
21.h odd 6 2 392.2.p.a 4
24.f even 2 1 224.2.b.a 2
24.h odd 2 1 56.2.b.a 2
48.i odd 4 2 1792.2.a.n 2
48.k even 4 2 1792.2.a.p 2
84.h odd 2 1 1568.2.b.a 2
84.j odd 6 2 1568.2.t.b 4
84.n even 6 2 1568.2.t.c 4
168.e odd 2 1 1568.2.b.a 2
168.i even 2 1 392.2.b.b 2
168.s odd 6 2 392.2.p.a 4
168.v even 6 2 1568.2.t.c 4
168.ba even 6 2 392.2.p.b 4
168.be odd 6 2 1568.2.t.b 4

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

 $$T_{5}^{2} + 2$$ $$T_{11}^{2} + 8$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2}$$
$3$ 
$5$ $$1 - 8 T^{2} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$13$ $$1 - 8 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 20 T^{2} + 361 T^{4}$$
$23$ $$( 1 - 6 T + 23 T^{2} )^{2}$$
$29$ $$1 - 50 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 2 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} )$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 74 T^{2} + 2809 T^{4}$$
$59$ $$1 - 116 T^{2} + 3481 T^{4}$$
$61$ $$1 + 40 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 67 T^{2} )^{2}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 2 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 8 T + 79 T^{2} )^{2}$$
$83$ $$1 + 76 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$