# Properties

 Label 648.2.a.d Level $648$ Weight $2$ Character orbit 648.a Self dual yes Analytic conductor $5.174$ Analytic rank $0$ 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{5} + O(q^{10})$$ $$q + q^{5} + 4q^{11} - 5q^{13} + 5q^{17} + 8q^{19} + 4q^{23} - 4q^{25} - 3q^{29} - 4q^{31} + 3q^{37} + 6q^{41} + 4q^{43} + 12q^{47} - 7q^{49} + 10q^{53} + 4q^{55} - 8q^{59} - 5q^{61} - 5q^{65} + 8q^{67} - 16q^{71} - 5q^{73} + 4q^{79} - 4q^{83} + 5q^{85} - 3q^{89} + 8q^{95} + 2q^{97} + O(q^{100})$$

## 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 0 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.d yes 1
3.b odd 2 1 648.2.a.b 1
4.b odd 2 1 1296.2.a.h 1
8.b even 2 1 5184.2.a.k 1
8.d odd 2 1 5184.2.a.l 1
9.c even 3 2 648.2.i.d 2
9.d odd 6 2 648.2.i.f 2
12.b even 2 1 1296.2.a.d 1
24.f even 2 1 5184.2.a.u 1
24.h odd 2 1 5184.2.a.v 1
36.f odd 6 2 1296.2.i.f 2
36.h even 6 2 1296.2.i.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.a.b 1 3.b odd 2 1
648.2.a.d yes 1 1.a even 1 1 trivial
648.2.i.d 2 9.c even 3 2
648.2.i.f 2 9.d odd 6 2
1296.2.a.d 1 12.b even 2 1
1296.2.a.h 1 4.b odd 2 1
1296.2.i.f 2 36.f odd 6 2
1296.2.i.k 2 36.h even 6 2
5184.2.a.k 1 8.b even 2 1
5184.2.a.l 1 8.d odd 2 1
5184.2.a.u 1 24.f even 2 1
5184.2.a.v 1 24.h odd 2 1

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{7}$$

## Hecke characteristic polynomials

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