# Properties

 Label 648.4.a.d Level $648$ Weight $4$ Character orbit 648.a Self dual yes Analytic conductor $38.233$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$38.2332376837$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{129})$$ Defining polynomial: $$x^{2} - x - 32$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{129}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta ) q^{5} + ( -3 - \beta ) q^{7} +O(q^{10})$$ $$q + ( -2 - \beta ) q^{5} + ( -3 - \beta ) q^{7} + ( -5 - 3 \beta ) q^{11} + ( -20 - 5 \beta ) q^{13} + ( 17 + 4 \beta ) q^{17} + ( 17 - 5 \beta ) q^{19} + ( 49 + 7 \beta ) q^{23} + ( 8 + 4 \beta ) q^{25} + ( 60 - 11 \beta ) q^{29} + ( -100 + 8 \beta ) q^{31} + ( 135 + 5 \beta ) q^{35} + ( 240 + 7 \beta ) q^{37} + ( 96 - 10 \beta ) q^{41} + ( -167 - 25 \beta ) q^{43} + ( 150 + 38 \beta ) q^{47} + ( -205 + 6 \beta ) q^{49} + ( 34 - 32 \beta ) q^{53} + ( 397 + 11 \beta ) q^{55} + ( 310 - 6 \beta ) q^{59} + ( 100 + 35 \beta ) q^{61} + ( 685 + 30 \beta ) q^{65} + ( -703 + 11 \beta ) q^{67} + ( 695 - 3 \beta ) q^{71} + ( 901 - 18 \beta ) q^{73} + ( 402 + 14 \beta ) q^{77} + ( -167 + 79 \beta ) q^{79} + ( -250 - 74 \beta ) q^{83} + ( -550 - 25 \beta ) q^{85} + ( 45 + 24 \beta ) q^{89} + ( 705 + 35 \beta ) q^{91} + ( 611 - 7 \beta ) q^{95} + ( -100 - 122 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 6q^{7} - 10q^{11} - 40q^{13} + 34q^{17} + 34q^{19} + 98q^{23} + 16q^{25} + 120q^{29} - 200q^{31} + 270q^{35} + 480q^{37} + 192q^{41} - 334q^{43} + 300q^{47} - 410q^{49} + 68q^{53} + 794q^{55} + 620q^{59} + 200q^{61} + 1370q^{65} - 1406q^{67} + 1390q^{71} + 1802q^{73} + 804q^{77} - 334q^{79} - 500q^{83} - 1100q^{85} + 90q^{89} + 1410q^{91} + 1222q^{95} - 200q^{97} + O(q^{100})$$

## 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
 6.17891 −5.17891
0 0 0 −13.3578 0 −14.3578 0 0 0
1.2 0 0 0 9.35782 0 8.35782 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 648.4.a.d 2
3.b odd 2 1 648.4.a.e yes 2
4.b odd 2 1 1296.4.a.n 2
9.c even 3 2 648.4.i.q 4
9.d odd 6 2 648.4.i.p 4
12.b even 2 1 1296.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.4.a.d 2 1.a even 1 1 trivial
648.4.a.e yes 2 3.b odd 2 1
648.4.i.p 4 9.d odd 6 2
648.4.i.q 4 9.c even 3 2
1296.4.a.n 2 4.b odd 2 1
1296.4.a.p 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4 T_{5} - 125$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(648))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-125 + 4 T + T^{2}$$
$7$ $$-120 + 6 T + T^{2}$$
$11$ $$-1136 + 10 T + T^{2}$$
$13$ $$-2825 + 40 T + T^{2}$$
$17$ $$-1775 - 34 T + T^{2}$$
$19$ $$-2936 - 34 T + T^{2}$$
$23$ $$-3920 - 98 T + T^{2}$$
$29$ $$-12009 - 120 T + T^{2}$$
$31$ $$1744 + 200 T + T^{2}$$
$37$ $$51279 - 480 T + T^{2}$$
$41$ $$-3684 - 192 T + T^{2}$$
$43$ $$-52736 + 334 T + T^{2}$$
$47$ $$-163776 - 300 T + T^{2}$$
$53$ $$-130940 - 68 T + T^{2}$$
$59$ $$91456 - 620 T + T^{2}$$
$61$ $$-148025 - 200 T + T^{2}$$
$67$ $$478600 + 1406 T + T^{2}$$
$71$ $$481864 - 1390 T + T^{2}$$
$73$ $$770005 - 1802 T + T^{2}$$
$79$ $$-777200 + 334 T + T^{2}$$
$83$ $$-643904 + 500 T + T^{2}$$
$89$ $$-72279 - 90 T + T^{2}$$
$97$ $$-1910036 + 200 T + T^{2}$$