# Properties

 Label 504.1.l.a Level 504 Weight 1 Character orbit 504.l Self dual yes Analytic conductor 0.252 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -7, -56, 8 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$0.251528766367$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{2}, \sqrt{-7})$$ Artin image $D_4$ Artin field Galois closure of 4.0.3528.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - q^{7} + q^{8} - q^{14} + q^{16} - 2q^{23} - q^{25} - q^{28} + q^{32} - 2q^{46} + q^{49} - q^{50} - q^{56} + q^{64} + 2q^{71} - 2q^{79} - 2q^{92} + q^{98} + O(q^{100})$$

## 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
 0
1.00000 0 1.00000 0 0 −1.00000 1.00000 0 0
 $$n$$: 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 CM by $$\Q(\sqrt{-7})$$
8.b even 2 1 RM by $$\Q(\sqrt{2})$$
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.1.l.a 1
3.b odd 2 1 56.1.h.a 1
4.b odd 2 1 2016.1.l.a 1
7.b odd 2 1 CM 504.1.l.a 1
7.c even 3 2 3528.1.bw.a 2
7.d odd 6 2 3528.1.bw.a 2
8.b even 2 1 RM 504.1.l.a 1
8.d odd 2 1 2016.1.l.a 1
12.b even 2 1 224.1.h.a 1
15.d odd 2 1 1400.1.m.a 1
15.e even 4 2 1400.1.c.a 2
21.c even 2 1 56.1.h.a 1
21.g even 6 2 392.1.j.a 2
21.h odd 6 2 392.1.j.a 2
24.f even 2 1 224.1.h.a 1
24.h odd 2 1 56.1.h.a 1
28.d even 2 1 2016.1.l.a 1
48.i odd 4 2 1792.1.c.b 1
48.k even 4 2 1792.1.c.a 1
56.e even 2 1 2016.1.l.a 1
56.h odd 2 1 CM 504.1.l.a 1
56.j odd 6 2 3528.1.bw.a 2
56.p even 6 2 3528.1.bw.a 2
84.h odd 2 1 224.1.h.a 1
84.j odd 6 2 1568.1.n.a 2
84.n even 6 2 1568.1.n.a 2
105.g even 2 1 1400.1.m.a 1
105.k odd 4 2 1400.1.c.a 2
120.i odd 2 1 1400.1.m.a 1
120.w even 4 2 1400.1.c.a 2
168.e odd 2 1 224.1.h.a 1
168.i even 2 1 56.1.h.a 1
168.s odd 6 2 392.1.j.a 2
168.v even 6 2 1568.1.n.a 2
168.ba even 6 2 392.1.j.a 2
168.be odd 6 2 1568.1.n.a 2
336.v odd 4 2 1792.1.c.a 1
336.y even 4 2 1792.1.c.b 1
840.u even 2 1 1400.1.m.a 1
840.bp odd 4 2 1400.1.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 3.b odd 2 1
56.1.h.a 1 21.c even 2 1
56.1.h.a 1 24.h odd 2 1
56.1.h.a 1 168.i even 2 1
224.1.h.a 1 12.b even 2 1
224.1.h.a 1 24.f even 2 1
224.1.h.a 1 84.h odd 2 1
224.1.h.a 1 168.e odd 2 1
392.1.j.a 2 21.g even 6 2
392.1.j.a 2 21.h odd 6 2
392.1.j.a 2 168.s odd 6 2
392.1.j.a 2 168.ba even 6 2
504.1.l.a 1 1.a even 1 1 trivial
504.1.l.a 1 7.b odd 2 1 CM
504.1.l.a 1 8.b even 2 1 RM
504.1.l.a 1 56.h odd 2 1 CM
1400.1.c.a 2 15.e even 4 2
1400.1.c.a 2 105.k odd 4 2
1400.1.c.a 2 120.w even 4 2
1400.1.c.a 2 840.bp odd 4 2
1400.1.m.a 1 15.d odd 2 1
1400.1.m.a 1 105.g even 2 1
1400.1.m.a 1 120.i odd 2 1
1400.1.m.a 1 840.u even 2 1
1568.1.n.a 2 84.j odd 6 2
1568.1.n.a 2 84.n even 6 2
1568.1.n.a 2 168.v even 6 2
1568.1.n.a 2 168.be odd 6 2
1792.1.c.a 1 48.k even 4 2
1792.1.c.a 1 336.v odd 4 2
1792.1.c.b 1 48.i odd 4 2
1792.1.c.b 1 336.y even 4 2
2016.1.l.a 1 4.b odd 2 1
2016.1.l.a 1 8.d odd 2 1
2016.1.l.a 1 28.d even 2 1
2016.1.l.a 1 56.e even 2 1
3528.1.bw.a 2 7.c even 3 2
3528.1.bw.a 2 7.d odd 6 2
3528.1.bw.a 2 56.j odd 6 2
3528.1.bw.a 2 56.p even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}$$ acting on $$S_{1}^{\mathrm{new}}(504, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ 1
$5$ $$1 + T^{2}$$
$7$ $$1 + T$$
$11$ $$( 1 - T )( 1 + T )$$
$13$ $$1 + T^{2}$$
$17$ $$( 1 - T )( 1 + T )$$
$19$ $$1 + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 1 - T )( 1 + T )$$
$31$ $$( 1 - T )( 1 + T )$$
$37$ $$( 1 - T )( 1 + T )$$
$41$ $$( 1 - T )( 1 + T )$$
$43$ $$( 1 - T )( 1 + T )$$
$47$ $$( 1 - T )( 1 + T )$$
$53$ $$( 1 - T )( 1 + T )$$
$59$ $$1 + T^{2}$$
$61$ $$1 + T^{2}$$
$67$ $$( 1 - T )( 1 + T )$$
$71$ $$( 1 - T )^{2}$$
$73$ $$( 1 - T )( 1 + T )$$
$79$ $$( 1 + T )^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$( 1 - T )( 1 + T )$$
$97$ $$( 1 - T )( 1 + T )$$