# Properties

 Label 644.2.a.a Level $644$ Weight $2$ Character orbit 644.a Self dual yes Analytic conductor $5.142$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$644 = 2^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 644.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.14236589017$$ 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

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + q^7 - 2 * q^9 $$q - q^{3} + q^{7} - 2 q^{9} - 2 q^{11} - 3 q^{13} - q^{21} - q^{23} - 5 q^{25} + 5 q^{27} + q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} + 3 q^{39} - 7 q^{41} - 4 q^{43} + 3 q^{47} + q^{49} - 12 q^{53} + 4 q^{59} - 6 q^{61} - 2 q^{63} - 12 q^{67} + q^{69} + 13 q^{71} + 3 q^{73} + 5 q^{75} - 2 q^{77} + 4 q^{79} + q^{81} + 16 q^{83} - q^{87} + 4 q^{89} - 3 q^{91} + 5 q^{93} + 10 q^{97} + 4 q^{99}+O(q^{100})$$ q - q^3 + q^7 - 2 * q^9 - 2 * q^11 - 3 * q^13 - q^21 - q^23 - 5 * q^25 + 5 * q^27 + q^29 - 5 * q^31 + 2 * q^33 - 8 * q^37 + 3 * q^39 - 7 * q^41 - 4 * q^43 + 3 * q^47 + q^49 - 12 * q^53 + 4 * q^59 - 6 * q^61 - 2 * q^63 - 12 * q^67 + q^69 + 13 * q^71 + 3 * q^73 + 5 * q^75 - 2 * q^77 + 4 * q^79 + q^81 + 16 * q^83 - q^87 + 4 * q^89 - 3 * q^91 + 5 * q^93 + 10 * q^97 + 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
0 −1.00000 0 0 0 1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## 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 644.2.a.a 1
3.b odd 2 1 5796.2.a.e 1
4.b odd 2 1 2576.2.a.l 1
7.b odd 2 1 4508.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.a 1 1.a even 1 1 trivial
2576.2.a.l 1 4.b odd 2 1
4508.2.a.c 1 7.b odd 2 1
5796.2.a.e 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(644))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 2$$
$13$ $$T + 3$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T - 1$$
$31$ $$T + 5$$
$37$ $$T + 8$$
$41$ $$T + 7$$
$43$ $$T + 4$$
$47$ $$T - 3$$
$53$ $$T + 12$$
$59$ $$T - 4$$
$61$ $$T + 6$$
$67$ $$T + 12$$
$71$ $$T - 13$$
$73$ $$T - 3$$
$79$ $$T - 4$$
$83$ $$T - 16$$
$89$ $$T - 4$$
$97$ $$T - 10$$