# Properties

 Label 504.3.l.a Level 504 Weight 3 Character orbit 504.l Self dual yes Analytic conductor 13.733 Analytic rank 0 Dimension 2 CM discriminant -56 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7330053238$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + 6 \beta q^{5} -7 q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + 6 \beta q^{5} -7 q^{7} -8 q^{8} -12 \beta q^{10} + 18 \beta q^{13} + 14 q^{14} + 16 q^{16} -6 \beta q^{19} + 24 \beta q^{20} + 10 q^{23} + 47 q^{25} -36 \beta q^{26} -28 q^{28} -32 q^{32} -42 \beta q^{35} + 12 \beta q^{38} -48 \beta q^{40} -20 q^{46} + 49 q^{49} -94 q^{50} + 72 \beta q^{52} + 56 q^{56} + 54 \beta q^{59} -6 \beta q^{61} + 64 q^{64} + 216 q^{65} + 84 \beta q^{70} + 110 q^{71} -24 \beta q^{76} + 130 q^{79} + 96 \beta q^{80} -18 \beta q^{83} -126 \beta q^{91} + 40 q^{92} -72 q^{95} -98 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 14q^{7} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 14q^{7} - 16q^{8} + 28q^{14} + 32q^{16} + 20q^{23} + 94q^{25} - 56q^{28} - 64q^{32} - 40q^{46} + 98q^{49} - 188q^{50} + 112q^{56} + 128q^{64} + 432q^{65} + 220q^{71} + 260q^{79} + 80q^{92} - 144q^{95} - 196q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 −1.41421 1.41421
−2.00000 0 4.00000 −8.48528 0 −7.00000 −8.00000 0 16.9706
181.2 −2.00000 0 4.00000 8.48528 0 −7.00000 −8.00000 0 −16.9706
 $$n$$: 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
7.b odd 2 1 inner
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 504.3.l.a 2
3.b odd 2 1 56.3.h.c 2
4.b odd 2 1 2016.3.l.c 2
7.b odd 2 1 inner 504.3.l.a 2
8.b even 2 1 inner 504.3.l.a 2
8.d odd 2 1 2016.3.l.c 2
12.b even 2 1 224.3.h.c 2
21.c even 2 1 56.3.h.c 2
21.g even 6 2 392.3.j.a 4
21.h odd 6 2 392.3.j.a 4
24.f even 2 1 224.3.h.c 2
24.h odd 2 1 56.3.h.c 2
28.d even 2 1 2016.3.l.c 2
56.e even 2 1 2016.3.l.c 2
56.h odd 2 1 CM 504.3.l.a 2
84.h odd 2 1 224.3.h.c 2
168.e odd 2 1 224.3.h.c 2
168.i even 2 1 56.3.h.c 2
168.s odd 6 2 392.3.j.a 4
168.ba even 6 2 392.3.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.h.c 2 3.b odd 2 1
56.3.h.c 2 21.c even 2 1
56.3.h.c 2 24.h odd 2 1
56.3.h.c 2 168.i even 2 1
224.3.h.c 2 12.b even 2 1
224.3.h.c 2 24.f even 2 1
224.3.h.c 2 84.h odd 2 1
224.3.h.c 2 168.e odd 2 1
392.3.j.a 4 21.g even 6 2
392.3.j.a 4 21.h odd 6 2
392.3.j.a 4 168.s odd 6 2
392.3.j.a 4 168.ba even 6 2
504.3.l.a 2 1.a even 1 1 trivial
504.3.l.a 2 7.b odd 2 1 inner
504.3.l.a 2 8.b even 2 1 inner
504.3.l.a 2 56.h odd 2 1 CM
2016.3.l.c 2 4.b odd 2 1
2016.3.l.c 2 8.d odd 2 1
2016.3.l.c 2 28.d even 2 1
2016.3.l.c 2 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(504, [\chi])$$:

 $$T_{5}^{2} - 72$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ 1
$5$ $$1 - 22 T^{2} + 625 T^{4}$$
$7$ $$( 1 + 7 T )^{2}$$
$11$ $$( 1 - 11 T )^{2}( 1 + 11 T )^{2}$$
$13$ $$1 - 310 T^{2} + 28561 T^{4}$$
$17$ $$( 1 - 17 T )^{2}( 1 + 17 T )^{2}$$
$19$ $$1 + 650 T^{2} + 130321 T^{4}$$
$23$ $$( 1 - 10 T + 529 T^{2} )^{2}$$
$29$ $$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$
$31$ $$( 1 - 31 T )^{2}( 1 + 31 T )^{2}$$
$37$ $$( 1 - 37 T )^{2}( 1 + 37 T )^{2}$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$( 1 - 43 T )^{2}( 1 + 43 T )^{2}$$
$47$ $$( 1 - 47 T )^{2}( 1 + 47 T )^{2}$$
$53$ $$( 1 - 53 T )^{2}( 1 + 53 T )^{2}$$
$59$ $$1 + 1130 T^{2} + 12117361 T^{4}$$
$61$ $$1 + 7370 T^{2} + 13845841 T^{4}$$
$67$ $$( 1 - 67 T )^{2}( 1 + 67 T )^{2}$$
$71$ $$( 1 - 110 T + 5041 T^{2} )^{2}$$
$73$ $$( 1 - 73 T )^{2}( 1 + 73 T )^{2}$$
$79$ $$( 1 - 130 T + 6241 T^{2} )^{2}$$
$83$ $$1 + 13130 T^{2} + 47458321 T^{4}$$
$89$ $$( 1 - 89 T )^{2}( 1 + 89 T )^{2}$$
$97$ $$( 1 - 97 T )^{2}( 1 + 97 T )^{2}$$