# Properties

 Label 663.2.a.a Level $663$ Weight $2$ Character orbit 663.a Self dual yes Analytic conductor $5.294$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 - 2 * q^5 + q^6 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + q^{13} + 2 q^{15} - q^{16} + q^{17} - q^{18} - 4 q^{19} + 2 q^{20} - 4 q^{22} - 3 q^{24} - q^{25} - q^{26} - q^{27} - 2 q^{29} - 2 q^{30} - 8 q^{31} - 5 q^{32} - 4 q^{33} - q^{34} - q^{36} - 2 q^{37} + 4 q^{38} - q^{39} - 6 q^{40} + 2 q^{41} - 4 q^{43} - 4 q^{44} - 2 q^{45} + 8 q^{47} + q^{48} - 7 q^{49} + q^{50} - q^{51} - q^{52} - 10 q^{53} + q^{54} - 8 q^{55} + 4 q^{57} + 2 q^{58} + 4 q^{59} - 2 q^{60} + 14 q^{61} + 8 q^{62} + 7 q^{64} - 2 q^{65} + 4 q^{66} - 4 q^{67} - q^{68} + 3 q^{72} - 14 q^{73} + 2 q^{74} + q^{75} + 4 q^{76} + q^{78} - 8 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} - 4 q^{83} - 2 q^{85} + 4 q^{86} + 2 q^{87} + 12 q^{88} - 6 q^{89} + 2 q^{90} + 8 q^{93} - 8 q^{94} + 8 q^{95} + 5 q^{96} - 6 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 - q^4 - 2 * q^5 + q^6 + 3 * q^8 + q^9 + 2 * q^10 + 4 * q^11 + q^12 + q^13 + 2 * q^15 - q^16 + q^17 - q^18 - 4 * q^19 + 2 * q^20 - 4 * q^22 - 3 * q^24 - q^25 - q^26 - q^27 - 2 * q^29 - 2 * q^30 - 8 * q^31 - 5 * q^32 - 4 * q^33 - q^34 - q^36 - 2 * q^37 + 4 * q^38 - q^39 - 6 * q^40 + 2 * q^41 - 4 * q^43 - 4 * q^44 - 2 * q^45 + 8 * q^47 + q^48 - 7 * q^49 + q^50 - q^51 - q^52 - 10 * q^53 + q^54 - 8 * q^55 + 4 * q^57 + 2 * q^58 + 4 * q^59 - 2 * q^60 + 14 * q^61 + 8 * q^62 + 7 * q^64 - 2 * q^65 + 4 * q^66 - 4 * q^67 - q^68 + 3 * q^72 - 14 * q^73 + 2 * q^74 + q^75 + 4 * q^76 + q^78 - 8 * q^79 + 2 * q^80 + q^81 - 2 * q^82 - 4 * q^83 - 2 * q^85 + 4 * q^86 + 2 * q^87 + 12 * q^88 - 6 * q^89 + 2 * q^90 + 8 * q^93 - 8 * q^94 + 8 * q^95 + 5 * q^96 - 6 * q^97 + 7 * q^98 + 4 * q^99

## 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 −2.00000 1.00000 0 3.00000 1.00000 2.00000
 $$n$$: 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.a 1
3.b odd 2 1 1989.2.a.e 1
13.b even 2 1 8619.2.a.i 1

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

 $$T_{2} + 1$$ T2 + 1 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T - 1$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T - 14$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 14$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T + 6$$
$97$ $$T + 6$$