# Properties

 Label 663.2.a.c Level $663$ Weight $2$ Character orbit 663.a Self dual yes Analytic conductor $5.294$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$663 = 3 \cdot 13 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 663.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.29408165401$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} - 3q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} - q^{4} - 4q^{5} - q^{6} + 2q^{7} - 3q^{8} + q^{9} - 4q^{10} + 6q^{11} + q^{12} - q^{13} + 2q^{14} + 4q^{15} - q^{16} + q^{17} + q^{18} + 4q^{19} + 4q^{20} - 2q^{21} + 6q^{22} + 3q^{24} + 11q^{25} - q^{26} - q^{27} - 2q^{28} - 6q^{29} + 4q^{30} + 10q^{31} + 5q^{32} - 6q^{33} + q^{34} - 8q^{35} - q^{36} - 4q^{37} + 4q^{38} + q^{39} + 12q^{40} - 2q^{42} + 12q^{43} - 6q^{44} - 4q^{45} + 8q^{47} + q^{48} - 3q^{49} + 11q^{50} - q^{51} + q^{52} + 2q^{53} - q^{54} - 24q^{55} - 6q^{56} - 4q^{57} - 6q^{58} - 8q^{59} - 4q^{60} - 10q^{61} + 10q^{62} + 2q^{63} + 7q^{64} + 4q^{65} - 6q^{66} + 12q^{67} - q^{68} - 8q^{70} + 2q^{71} - 3q^{72} + 4q^{73} - 4q^{74} - 11q^{75} - 4q^{76} + 12q^{77} + q^{78} - 4q^{79} + 4q^{80} + q^{81} - 12q^{83} + 2q^{84} - 4q^{85} + 12q^{86} + 6q^{87} - 18q^{88} - 6q^{89} - 4q^{90} - 2q^{91} - 10q^{93} + 8q^{94} - 16q^{95} - 5q^{96} + 8q^{97} - 3q^{98} + 6q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 −1.00000 −1.00000 −4.00000 −1.00000 2.00000 −3.00000 1.00000 −4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$13$$ $$1$$
$$17$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.a.c 1
3.b odd 2 1 1989.2.a.b 1
13.b even 2 1 8619.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.a.c 1 1.a even 1 1 trivial
1989.2.a.b 1 3.b odd 2 1
8619.2.a.e 1 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(663))$$:

 $$T_{2} - 1$$ $$T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$4 + T$$
$7$ $$-2 + T$$
$11$ $$-6 + T$$
$13$ $$1 + T$$
$17$ $$-1 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-10 + T$$
$37$ $$4 + T$$
$41$ $$T$$
$43$ $$-12 + T$$
$47$ $$-8 + T$$
$53$ $$-2 + T$$
$59$ $$8 + T$$
$61$ $$10 + T$$
$67$ $$-12 + T$$
$71$ $$-2 + T$$
$73$ $$-4 + T$$
$79$ $$4 + T$$
$83$ $$12 + T$$
$89$ $$6 + T$$
$97$ $$-8 + T$$