# Properties

 Label 3648.2.a.bx Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.1294266574$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 Defining polynomial: $$x^{3} - x^{2} - 5 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1824) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -2 + \beta_{2} ) q^{7} + q^{9} + ( -2 - \beta_{2} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + \beta_{2} q^{17} - q^{19} + ( 2 - \beta_{2} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} ) q^{23} + ( 5 - \beta_{2} ) q^{25} - q^{27} + ( -2 + 2 \beta_{2} ) q^{29} + ( -3 - \beta_{1} + \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{33} + ( -4 + 2 \beta_{1} + 3 \beta_{2} ) q^{35} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} ) q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 3 - \beta_{1} + 2 \beta_{2} ) q^{47} + ( 5 - 2 \beta_{1} - 3 \beta_{2} ) q^{49} -\beta_{2} q^{51} + 2 q^{53} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{55} + q^{57} + ( -4 + 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{61} + ( -2 + \beta_{2} ) q^{63} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 2 - 2 \beta_{1} ) q^{67} + ( -1 + \beta_{1} + \beta_{2} ) q^{69} + ( 4 + 2 \beta_{2} ) q^{71} + ( 12 + \beta_{2} ) q^{73} + ( -5 + \beta_{2} ) q^{75} + ( -4 + 2 \beta_{1} - \beta_{2} ) q^{77} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 4 - 4 \beta_{2} ) q^{83} + ( -2 + 3 \beta_{2} ) q^{85} + ( 2 - 2 \beta_{2} ) q^{87} + 10 q^{89} + 8 q^{91} + ( 3 + \beta_{1} - \beta_{2} ) q^{93} + ( -1 + \beta_{1} ) q^{95} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 3q^{5} - 5q^{7} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 3q^{5} - 5q^{7} + 3q^{9} - 7q^{11} - 2q^{13} - 3q^{15} + q^{17} - 3q^{19} + 5q^{21} + 2q^{23} + 14q^{25} - 3q^{27} - 4q^{29} - 8q^{31} + 7q^{33} - 9q^{35} - 6q^{37} + 2q^{39} + 12q^{41} + q^{43} + 3q^{45} + 11q^{47} + 12q^{49} - q^{51} + 6q^{53} - 3q^{55} + 3q^{57} - 10q^{59} - 7q^{61} - 5q^{63} + 20q^{65} + 6q^{67} - 2q^{69} + 14q^{71} + 37q^{73} - 14q^{75} - 13q^{77} - 4q^{79} + 3q^{81} + 8q^{83} - 3q^{85} + 4q^{87} + 30q^{89} + 24q^{91} + 8q^{93} - 3q^{95} + 8q^{97} - 7q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 7$$$$)/2$$

## 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
 2.39138 −2.16425 0.772866
0 −1.00000 0 −3.11009 0 −1.67267 0 1.00000 0
1.2 0 −1.00000 0 2.48028 0 1.84822 0 1.00000 0
1.3 0 −1.00000 0 3.62981 0 −5.17554 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$1$$

## 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 3648.2.a.bx 3
4.b odd 2 1 3648.2.a.bz 3
8.b even 2 1 1824.2.a.u yes 3
8.d odd 2 1 1824.2.a.s 3
24.f even 2 1 5472.2.a.bp 3
24.h odd 2 1 5472.2.a.bo 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.s 3 8.d odd 2 1
1824.2.a.u yes 3 8.b even 2 1
3648.2.a.bx 3 1.a even 1 1 trivial
3648.2.a.bz 3 4.b odd 2 1
5472.2.a.bo 3 24.h odd 2 1
5472.2.a.bp 3 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5}^{3} - 3 T_{5}^{2} - 10 T_{5} + 28$$ $$T_{7}^{3} + 5 T_{7}^{2} - 4 T_{7} - 16$$ $$T_{11}^{3} + 7 T_{11}^{2} + 4 T_{11} - 16$$ $$T_{23}^{3} - 2 T_{23}^{2} - 28 T_{23} - 32$$ $$T_{31}^{3} + 8 T_{31}^{2} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$28 - 10 T - 3 T^{2} + T^{3}$$
$7$ $$-16 - 4 T + 5 T^{2} + T^{3}$$
$11$ $$-16 + 4 T + 7 T^{2} + T^{3}$$
$13$ $$-32 - 20 T + 2 T^{2} + T^{3}$$
$17$ $$4 - 12 T - T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$-32 - 28 T - 2 T^{2} + T^{3}$$
$29$ $$-64 - 44 T + 4 T^{2} + T^{3}$$
$31$ $$-56 + 8 T^{2} + T^{3}$$
$37$ $$-752 - 124 T + 6 T^{2} + T^{3}$$
$41$ $$272 - 4 T - 12 T^{2} + T^{3}$$
$43$ $$112 - 56 T - T^{2} + T^{3}$$
$47$ $$-4 - 14 T - 11 T^{2} + T^{3}$$
$53$ $$( -2 + T )^{3}$$
$59$ $$-128 - 16 T + 10 T^{2} + T^{3}$$
$61$ $$-212 - 40 T + 7 T^{2} + T^{3}$$
$67$ $$224 - 40 T - 6 T^{2} + T^{3}$$
$71$ $$128 + 16 T - 14 T^{2} + T^{3}$$
$73$ $$-1724 + 444 T - 37 T^{2} + T^{3}$$
$79$ $$-56 - 136 T + 4 T^{2} + T^{3}$$
$83$ $$512 - 176 T - 8 T^{2} + T^{3}$$
$89$ $$( -10 + T )^{3}$$
$97$ $$1792 - 196 T - 8 T^{2} + T^{3}$$