# Properties

 Label 648.2.a.g Level $648$ Weight $2$ Character orbit 648.a Self dual yes Analytic conductor $5.174$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} + ( 2 - \beta ) q^{7} +O(q^{10})$$ $$q + \beta q^{5} + ( 2 - \beta ) q^{7} - q^{11} + ( 2 + \beta ) q^{13} + ( 3 - \beta ) q^{17} + ( 3 + \beta ) q^{19} + ( -2 - \beta ) q^{23} + ( 3 + \beta ) q^{25} + ( 2 - \beta ) q^{29} + ( 4 - \beta ) q^{31} + ( -8 + \beta ) q^{35} + ( 4 - 2 \beta ) q^{37} + ( 7 - 2 \beta ) q^{41} + ( 3 + 2 \beta ) q^{43} + ( -2 + \beta ) q^{47} + ( 5 - 3 \beta ) q^{49} + ( 4 + 2 \beta ) q^{53} -\beta q^{55} -7 q^{59} -\beta q^{61} + ( 8 + 3 \beta ) q^{65} + ( 1 + 2 \beta ) q^{67} + 4 q^{71} + ( -5 + 3 \beta ) q^{73} + ( -2 + \beta ) q^{77} + ( -4 + \beta ) q^{79} + ( -12 - \beta ) q^{83} + ( -8 + 2 \beta ) q^{85} -6 q^{89} + ( -4 - \beta ) q^{91} + ( 8 + 4 \beta ) q^{95} + ( -3 - 2 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{5} + 3q^{7} + O(q^{10})$$ $$2q + q^{5} + 3q^{7} - 2q^{11} + 5q^{13} + 5q^{17} + 7q^{19} - 5q^{23} + 7q^{25} + 3q^{29} + 7q^{31} - 15q^{35} + 6q^{37} + 12q^{41} + 8q^{43} - 3q^{47} + 7q^{49} + 10q^{53} - q^{55} - 14q^{59} - q^{61} + 19q^{65} + 4q^{67} + 8q^{71} - 7q^{73} - 3q^{77} - 7q^{79} - 25q^{83} - 14q^{85} - 12q^{89} - 9q^{91} + 20q^{95} - 8q^{97} + O(q^{100})$$

## 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
 −2.37228 3.37228
0 0 0 −2.37228 0 4.37228 0 0 0
1.2 0 0 0 3.37228 0 −1.37228 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.g 2
3.b odd 2 1 648.2.a.f 2
4.b odd 2 1 1296.2.a.p 2
8.b even 2 1 5184.2.a.bp 2
8.d odd 2 1 5184.2.a.bo 2
9.c even 3 2 216.2.i.b 4
9.d odd 6 2 72.2.i.b 4
12.b even 2 1 1296.2.a.n 2
24.f even 2 1 5184.2.a.bs 2
24.h odd 2 1 5184.2.a.bt 2
36.f odd 6 2 432.2.i.d 4
36.h even 6 2 144.2.i.d 4
72.j odd 6 2 576.2.i.j 4
72.l even 6 2 576.2.i.l 4
72.n even 6 2 1728.2.i.i 4
72.p odd 6 2 1728.2.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 9.d odd 6 2
144.2.i.d 4 36.h even 6 2
216.2.i.b 4 9.c even 3 2
432.2.i.d 4 36.f odd 6 2
576.2.i.j 4 72.j odd 6 2
576.2.i.l 4 72.l even 6 2
648.2.a.f 2 3.b odd 2 1
648.2.a.g 2 1.a even 1 1 trivial
1296.2.a.n 2 12.b even 2 1
1296.2.a.p 2 4.b odd 2 1
1728.2.i.i 4 72.n even 6 2
1728.2.i.j 4 72.p odd 6 2
5184.2.a.bo 2 8.d odd 2 1
5184.2.a.bp 2 8.b even 2 1
5184.2.a.bs 2 24.f even 2 1
5184.2.a.bt 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(648))$$:

 $$T_{5}^{2} - T_{5} - 8$$ $$T_{7}^{2} - 3 T_{7} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 - T + T^{2}$$
$7$ $$-6 - 3 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-2 - 5 T + T^{2}$$
$17$ $$-2 - 5 T + T^{2}$$
$19$ $$4 - 7 T + T^{2}$$
$23$ $$-2 + 5 T + T^{2}$$
$29$ $$-6 - 3 T + T^{2}$$
$31$ $$4 - 7 T + T^{2}$$
$37$ $$-24 - 6 T + T^{2}$$
$41$ $$3 - 12 T + T^{2}$$
$43$ $$-17 - 8 T + T^{2}$$
$47$ $$-6 + 3 T + T^{2}$$
$53$ $$-8 - 10 T + T^{2}$$
$59$ $$( 7 + T )^{2}$$
$61$ $$-8 + T + T^{2}$$
$67$ $$-29 - 4 T + T^{2}$$
$71$ $$( -4 + T )^{2}$$
$73$ $$-62 + 7 T + T^{2}$$
$79$ $$4 + 7 T + T^{2}$$
$83$ $$148 + 25 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$-17 + 8 T + T^{2}$$