# Properties

 Label 648.2.a.a Level $648$ Weight $2$ Character orbit 648.a Self dual yes Analytic conductor $5.174$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.17430605098$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} - 3q^{7} + O(q^{10})$$ $$q - q^{5} - 3q^{7} + 5q^{11} - 5q^{13} - 2q^{17} - 4q^{19} - q^{23} - 4q^{25} - 9q^{29} - q^{31} + 3q^{35} - 6q^{37} + 3q^{41} + q^{43} - 3q^{47} + 2q^{49} + 2q^{53} - 5q^{55} + 11q^{59} + 7q^{61} + 5q^{65} - q^{67} + 4q^{71} - 2q^{73} - 15q^{77} + q^{79} + q^{83} + 2q^{85} - 18q^{89} + 15q^{91} + 4q^{95} - 13q^{97} + O(q^{100})$$

## 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
0 0 0 −1.00000 0 −3.00000 0 0 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
$$2$$ $$1$$
$$3$$ $$1$$

## 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 648.2.a.a 1
3.b odd 2 1 648.2.a.c 1
4.b odd 2 1 1296.2.a.e 1
8.b even 2 1 5184.2.a.s 1
8.d odd 2 1 5184.2.a.x 1
9.c even 3 2 72.2.i.a 2
9.d odd 6 2 216.2.i.a 2
12.b even 2 1 1296.2.a.i 1
24.f even 2 1 5184.2.a.n 1
24.h odd 2 1 5184.2.a.i 1
36.f odd 6 2 144.2.i.b 2
36.h even 6 2 432.2.i.a 2
72.j odd 6 2 1728.2.i.h 2
72.l even 6 2 1728.2.i.g 2
72.n even 6 2 576.2.i.d 2
72.p odd 6 2 576.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 9.c even 3 2
144.2.i.b 2 36.f odd 6 2
216.2.i.a 2 9.d odd 6 2
432.2.i.a 2 36.h even 6 2
576.2.i.c 2 72.p odd 6 2
576.2.i.d 2 72.n even 6 2
648.2.a.a 1 1.a even 1 1 trivial
648.2.a.c 1 3.b odd 2 1
1296.2.a.e 1 4.b odd 2 1
1296.2.a.i 1 12.b even 2 1
1728.2.i.g 2 72.l even 6 2
1728.2.i.h 2 72.j odd 6 2
5184.2.a.i 1 24.h odd 2 1
5184.2.a.n 1 24.f even 2 1
5184.2.a.s 1 8.b even 2 1
5184.2.a.x 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(648))$$:

 $$T_{5} + 1$$ $$T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$3 + T$$
$11$ $$-5 + T$$
$13$ $$5 + T$$
$17$ $$2 + T$$
$19$ $$4 + T$$
$23$ $$1 + T$$
$29$ $$9 + T$$
$31$ $$1 + T$$
$37$ $$6 + T$$
$41$ $$-3 + T$$
$43$ $$-1 + T$$
$47$ $$3 + T$$
$53$ $$-2 + T$$
$59$ $$-11 + T$$
$61$ $$-7 + T$$
$67$ $$1 + T$$
$71$ $$-4 + T$$
$73$ $$2 + T$$
$79$ $$-1 + T$$
$83$ $$-1 + T$$
$89$ $$18 + T$$
$97$ $$13 + T$$