# Properties

 Label 663.1.g.a Level $663$ Weight $1$ Character orbit 663.g Self dual yes Analytic conductor $0.331$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -51, -663, 13 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$663 = 3 \cdot 13 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 663.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.330880103376$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{13}, \sqrt{-51})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.8619.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{4} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{4} + q^{9} + q^{12} + q^{13} + q^{16} - q^{17} + 2q^{23} + q^{25} - q^{27} + 2q^{29} - q^{36} - q^{39} - 2q^{43} - q^{48} - q^{49} + q^{51} - q^{52} - q^{64} + q^{68} - 2q^{69} - q^{75} + q^{81} - 2q^{87} - 2q^{92} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/663\mathbb{Z}\right)^\times$$.

 $$n$$ $$443$$ $$547$$ $$613$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
662.1
 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
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
51.c odd 2 1 CM by $$\Q(\sqrt{-51})$$
663.g odd 2 1 CM by $$\Q(\sqrt{-663})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.1.g.a 1
3.b odd 2 1 663.1.g.b yes 1
13.b even 2 1 RM 663.1.g.a 1
17.b even 2 1 663.1.g.b yes 1
39.d odd 2 1 663.1.g.b yes 1
51.c odd 2 1 CM 663.1.g.a 1
221.b even 2 1 663.1.g.b yes 1
663.g odd 2 1 CM 663.1.g.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.1.g.a 1 1.a even 1 1 trivial
663.1.g.a 1 13.b even 2 1 RM
663.1.g.a 1 51.c odd 2 1 CM
663.1.g.a 1 663.g odd 2 1 CM
663.1.g.b yes 1 3.b odd 2 1
663.1.g.b yes 1 17.b even 2 1
663.1.g.b yes 1 39.d odd 2 1
663.1.g.b yes 1 221.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(663, [\chi])$$:

 $$T_{2}$$ $$T_{23} - 2$$ $$T_{107} + 2$$

## Hecke characteristic polynomials

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