# Properties

 Label 507.1.c.a Level $507$ Weight $1$ Character orbit 507.c Self dual yes Analytic conductor $0.253$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 507.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.253025961405$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.6591.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{4} + q^{9} - q^{12} + q^{16} + q^{25} - q^{27} + q^{36} - 2q^{43} - q^{48} - q^{49} - 2q^{61} + q^{64} - q^{75} - 2q^{79} + q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1
 0
0 −1.00000 1.00000 0 0 0 0 1.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.c.a 1
3.b odd 2 1 CM 507.1.c.a 1
13.b even 2 1 RM 507.1.c.a 1
13.c even 3 2 507.1.i.a 2
13.d odd 4 2 39.1.d.a 1
13.e even 6 2 507.1.i.a 2
13.f odd 12 4 507.1.h.a 2
39.d odd 2 1 CM 507.1.c.a 1
39.f even 4 2 39.1.d.a 1
39.h odd 6 2 507.1.i.a 2
39.i odd 6 2 507.1.i.a 2
39.k even 12 4 507.1.h.a 2
52.f even 4 2 624.1.l.a 1
65.f even 4 2 975.1.e.a 2
65.g odd 4 2 975.1.g.a 1
65.k even 4 2 975.1.e.a 2
91.i even 4 2 1911.1.h.a 1
91.z odd 12 4 1911.1.w.b 2
91.bb even 12 4 1911.1.w.a 2
104.j odd 4 2 2496.1.l.b 1
104.m even 4 2 2496.1.l.a 1
117.y odd 12 4 1053.1.n.b 2
117.z even 12 4 1053.1.n.b 2
156.l odd 4 2 624.1.l.a 1
195.j odd 4 2 975.1.e.a 2
195.n even 4 2 975.1.g.a 1
195.u odd 4 2 975.1.e.a 2
273.o odd 4 2 1911.1.h.a 1
273.cb odd 12 4 1911.1.w.a 2
273.cd even 12 4 1911.1.w.b 2
312.w odd 4 2 2496.1.l.a 1
312.y even 4 2 2496.1.l.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 13.d odd 4 2
39.1.d.a 1 39.f even 4 2
507.1.c.a 1 1.a even 1 1 trivial
507.1.c.a 1 3.b odd 2 1 CM
507.1.c.a 1 13.b even 2 1 RM
507.1.c.a 1 39.d odd 2 1 CM
507.1.h.a 2 13.f odd 12 4
507.1.h.a 2 39.k even 12 4
507.1.i.a 2 13.c even 3 2
507.1.i.a 2 13.e even 6 2
507.1.i.a 2 39.h odd 6 2
507.1.i.a 2 39.i odd 6 2
624.1.l.a 1 52.f even 4 2
624.1.l.a 1 156.l odd 4 2
975.1.e.a 2 65.f even 4 2
975.1.e.a 2 65.k even 4 2
975.1.e.a 2 195.j odd 4 2
975.1.e.a 2 195.u odd 4 2
975.1.g.a 1 65.g odd 4 2
975.1.g.a 1 195.n even 4 2
1053.1.n.b 2 117.y odd 12 4
1053.1.n.b 2 117.z even 12 4
1911.1.h.a 1 91.i even 4 2
1911.1.h.a 1 273.o odd 4 2
1911.1.w.a 2 91.bb even 12 4
1911.1.w.a 2 273.cb odd 12 4
1911.1.w.b 2 91.z odd 12 4
1911.1.w.b 2 273.cd even 12 4
2496.1.l.a 1 104.m even 4 2
2496.1.l.a 1 312.w odd 4 2
2496.1.l.b 1 104.j odd 4 2
2496.1.l.b 1 312.y even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(507, [\chi])$$.

## Hecke characteristic polynomials

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