# Properties

 Label 648.2.a.h 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{5} + 2 \beta q^{7} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{5} + 2 \beta q^{7} + 2 q^{11} + ( 1 - 2 \beta ) q^{13} + ( 4 - \beta ) q^{17} + ( -4 - 2 \beta ) q^{19} + ( 2 - 4 \beta ) q^{23} + ( 2 + 4 \beta ) q^{25} + ( 6 - \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + ( 6 + 4 \beta ) q^{35} + ( 3 - 2 \beta ) q^{37} + 4 \beta q^{41} + ( -8 + 2 \beta ) q^{43} + 4 \beta q^{47} + 5 q^{49} + ( -4 - 4 \beta ) q^{53} + ( 4 + 2 \beta ) q^{55} + 8 q^{59} + ( 7 + 2 \beta ) q^{61} + ( -4 - 3 \beta ) q^{65} + ( -4 + 2 \beta ) q^{67} -2 q^{71} + q^{73} + 4 \beta q^{77} + ( 4 - 2 \beta ) q^{79} + ( 4 - 4 \beta ) q^{83} + ( 5 + 2 \beta ) q^{85} -3 \beta q^{89} + ( -12 + 2 \beta ) q^{91} + ( -14 - 8 \beta ) q^{95} + ( 2 - 8 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 4q^{11} + 2q^{13} + 8q^{17} - 8q^{19} + 4q^{23} + 4q^{25} + 12q^{29} - 8q^{31} + 12q^{35} + 6q^{37} - 16q^{43} + 10q^{49} - 8q^{53} + 8q^{55} + 16q^{59} + 14q^{61} - 8q^{65} - 8q^{67} - 4q^{71} + 2q^{73} + 8q^{79} + 8q^{83} + 10q^{85} - 24q^{91} - 28q^{95} + 4q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 0 0 0.267949 0 −3.46410 0 0 0
1.2 0 0 0 3.73205 0 3.46410 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.h yes 2
3.b odd 2 1 648.2.a.e 2
4.b odd 2 1 1296.2.a.q 2
8.b even 2 1 5184.2.a.bg 2
8.d odd 2 1 5184.2.a.bi 2
9.c even 3 2 648.2.i.i 4
9.d odd 6 2 648.2.i.j 4
12.b even 2 1 1296.2.a.m 2
24.f even 2 1 5184.2.a.bz 2
24.h odd 2 1 5184.2.a.cb 2
36.f odd 6 2 1296.2.i.r 4
36.h even 6 2 1296.2.i.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.a.e 2 3.b odd 2 1
648.2.a.h yes 2 1.a even 1 1 trivial
648.2.i.i 4 9.c even 3 2
648.2.i.j 4 9.d odd 6 2
1296.2.a.m 2 12.b even 2 1
1296.2.a.q 2 4.b odd 2 1
1296.2.i.r 4 36.f odd 6 2
1296.2.i.t 4 36.h even 6 2
5184.2.a.bg 2 8.b even 2 1
5184.2.a.bi 2 8.d odd 2 1
5184.2.a.bz 2 24.f even 2 1
5184.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(648))$$:

 $$T_{5}^{2} - 4 T_{5} + 1$$ $$T_{7}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$1 - 4 T + T^{2}$$
$7$ $$-12 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-11 - 2 T + T^{2}$$
$17$ $$13 - 8 T + T^{2}$$
$19$ $$4 + 8 T + T^{2}$$
$23$ $$-44 - 4 T + T^{2}$$
$29$ $$33 - 12 T + T^{2}$$
$31$ $$-32 + 8 T + T^{2}$$
$37$ $$-3 - 6 T + T^{2}$$
$41$ $$-48 + T^{2}$$
$43$ $$52 + 16 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-32 + 8 T + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$37 - 14 T + T^{2}$$
$67$ $$4 + 8 T + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$( -1 + T )^{2}$$
$79$ $$4 - 8 T + T^{2}$$
$83$ $$-32 - 8 T + T^{2}$$
$89$ $$-27 + T^{2}$$
$97$ $$-188 - 4 T + T^{2}$$