# Properties

 Label 3864.2.a.o Level $3864$ Weight $2$ Character orbit 3864.a Self dual yes Analytic conductor $30.854$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$30.8541953410$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.733.1 Defining polynomial: $$x^{3} - x^{2} - 7 x + 8$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} - q^{7} + q^{9} + ( -1 + \beta_{1} + \beta_{2} ) q^{11} + ( 2 - \beta_{1} ) q^{13} + ( -1 - \beta_{2} ) q^{15} + ( -1 - \beta_{1} + \beta_{2} ) q^{17} + ( -5 + \beta_{1} + \beta_{2} ) q^{19} - q^{21} + q^{23} + ( 3 - \beta_{1} ) q^{25} + q^{27} + ( -1 - \beta_{1} + \beta_{2} ) q^{29} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{31} + ( -1 + \beta_{1} + \beta_{2} ) q^{33} + ( 1 + \beta_{2} ) q^{35} + ( -1 - 3 \beta_{1} - \beta_{2} ) q^{37} + ( 2 - \beta_{1} ) q^{39} + ( 3 + \beta_{1} + \beta_{2} ) q^{41} + ( -8 + 3 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -1 - \beta_{2} ) q^{45} -2 \beta_{1} q^{47} + q^{49} + ( -1 - \beta_{1} + \beta_{2} ) q^{51} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{53} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{55} + ( -5 + \beta_{1} + \beta_{2} ) q^{57} + ( 3 - 4 \beta_{1} - \beta_{2} ) q^{59} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{61} - q^{63} + ( -5 + 3 \beta_{1} - \beta_{2} ) q^{65} + ( -3 - 4 \beta_{1} - \beta_{2} ) q^{67} + q^{69} + \beta_{1} q^{71} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{73} + ( 3 - \beta_{1} ) q^{75} + ( 1 - \beta_{1} - \beta_{2} ) q^{77} + ( -5 - \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 1 + 3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -9 + 4 \beta_{1} + 3 \beta_{2} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} ) q^{87} + ( -2 - 5 \beta_{1} ) q^{89} + ( -2 + \beta_{1} ) q^{91} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{93} + ( 1 - 2 \beta_{1} + 5 \beta_{2} ) q^{95} + ( 3 - \beta_{1} + \beta_{2} ) q^{97} + ( -1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} - 2q^{11} + 5q^{13} - 3q^{15} - 4q^{17} - 14q^{19} - 3q^{21} + 3q^{23} + 8q^{25} + 3q^{27} - 4q^{29} - 6q^{31} - 2q^{33} + 3q^{35} - 6q^{37} + 5q^{39} + 10q^{41} - 21q^{43} - 3q^{45} - 2q^{47} + 3q^{49} - 4q^{51} - 13q^{53} - 11q^{55} - 14q^{57} + 5q^{59} - 17q^{61} - 3q^{63} - 12q^{65} - 13q^{67} + 3q^{69} + q^{71} + 6q^{73} + 8q^{75} + 2q^{77} - 16q^{79} + 3q^{81} + 6q^{83} - 23q^{85} - 4q^{87} - 11q^{89} - 5q^{91} - 6q^{93} + q^{95} + 8q^{97} - 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 7 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$

## 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
 −2.69639 2.51820 1.17819
0 1.00000 0 −3.27053 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −2.34132 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.61186 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 3864.2.a.o 3
4.b odd 2 1 7728.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.o 3 1.a even 1 1 trivial
7728.2.a.bq 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3864))$$:

 $$T_{5}^{3} + 3 T_{5}^{2} - 7 T_{5} - 20$$ $$T_{11}^{3} + 2 T_{11}^{2} - 9 T_{11} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-20 - 7 T + 3 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-14 - 9 T + 2 T^{2} + T^{3}$$
$13$ $$2 + T - 5 T^{2} + T^{3}$$
$17$ $$-50 - 19 T + 4 T^{2} + T^{3}$$
$19$ $$46 + 55 T + 14 T^{2} + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$-50 - 19 T + 4 T^{2} + T^{3}$$
$31$ $$-160 - 43 T + 6 T^{2} + T^{3}$$
$37$ $$-44 - 43 T + 6 T^{2} + T^{3}$$
$41$ $$-10 + 23 T - 10 T^{2} + T^{3}$$
$43$ $$-302 + 83 T + 21 T^{2} + T^{3}$$
$47$ $$-64 - 28 T + 2 T^{2} + T^{3}$$
$53$ $$-16 + 3 T + 13 T^{2} + T^{3}$$
$59$ $$184 - 91 T - 5 T^{2} + T^{3}$$
$61$ $$-196 + 35 T + 17 T^{2} + T^{3}$$
$67$ $$-326 - 43 T + 13 T^{2} + T^{3}$$
$71$ $$8 - 7 T - T^{2} + T^{3}$$
$73$ $$550 - 85 T - 6 T^{2} + T^{3}$$
$79$ $$104 + 75 T + 16 T^{2} + T^{3}$$
$83$ $$-22 - 81 T - 6 T^{2} + T^{3}$$
$89$ $$-1322 - 143 T + 11 T^{2} + T^{3}$$
$97$ $$26 - 3 T - 8 T^{2} + T^{3}$$