Properties

 Label 6384.2.a.bw Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6384.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$50.9764966504$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 3192) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta_{1} q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta_{1} q^{5} + q^{7} + q^{9} + ( 1 + \beta_{1} + \beta_{2} ) q^{11} + 4 q^{13} -\beta_{1} q^{15} -\beta_{1} q^{17} + q^{19} + q^{21} + ( 1 + \beta_{1} + \beta_{2} ) q^{23} + ( 2 - \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 1 + \beta_{2} ) q^{29} + ( 3 + \beta_{1} + \beta_{2} ) q^{31} + ( 1 + \beta_{1} + \beta_{2} ) q^{33} -\beta_{1} q^{35} + 2 q^{37} + 4 q^{39} -2 \beta_{2} q^{41} + ( 1 - \beta_{1} - \beta_{2} ) q^{43} -\beta_{1} q^{45} + ( 1 - \beta_{2} ) q^{47} + q^{49} -\beta_{1} q^{51} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{53} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{55} + q^{57} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{61} + q^{63} -4 \beta_{1} q^{65} + ( -1 - \beta_{1} - \beta_{2} ) q^{67} + ( 1 + \beta_{1} + \beta_{2} ) q^{69} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{71} + ( 4 - 2 \beta_{2} ) q^{73} + ( 2 - \beta_{1} + \beta_{2} ) q^{75} + ( 1 + \beta_{1} + \beta_{2} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( 7 - \beta_{1} + \beta_{2} ) q^{85} + ( 1 + \beta_{2} ) q^{87} + ( 4 + 2 \beta_{2} ) q^{89} + 4 q^{91} + ( 3 + \beta_{1} + \beta_{2} ) q^{93} -\beta_{1} q^{95} + ( 9 + \beta_{1} - \beta_{2} ) q^{97} + ( 1 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu - 4$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 3 \beta_{1} + 9$$$$)/2$$

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
 3.12489 −1.76156 −0.363328
0 1.00000 0 −2.64002 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.864641 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 3.50466 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

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 6384.2.a.bw 3
4.b odd 2 1 3192.2.a.u 3
12.b even 2 1 9576.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.u 3 4.b odd 2 1
6384.2.a.bw 3 1.a even 1 1 trivial
9576.2.a.cb 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{3} - 10 T_{5} - 8$$ $$T_{11}^{3} - 2 T_{11}^{2} - 24 T_{11} - 16$$ $$T_{13} - 4$$ $$T_{17}^{3} - 10 T_{17} - 8$$ $$T_{23}^{3} - 2 T_{23}^{2} - 24 T_{23} - 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-8 - 10 T + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$-16 - 24 T - 2 T^{2} + T^{3}$$
$13$ $$( -4 + T )^{3}$$
$17$ $$-8 - 10 T + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$-16 - 24 T - 2 T^{2} + T^{3}$$
$29$ $$44 - 18 T - 2 T^{2} + T^{3}$$
$31$ $$16 - 4 T - 8 T^{2} + T^{3}$$
$37$ $$( -2 + T )^{3}$$
$41$ $$-200 - 76 T - 2 T^{2} + T^{3}$$
$43$ $$64 - 20 T - 4 T^{2} + T^{3}$$
$47$ $$-8 - 14 T - 4 T^{2} + T^{3}$$
$53$ $$-4 + 14 T + 14 T^{2} + T^{3}$$
$59$ $$-256 - 128 T + 4 T^{2} + T^{3}$$
$61$ $$8 - 132 T - 2 T^{2} + T^{3}$$
$67$ $$16 - 24 T + 2 T^{2} + T^{3}$$
$71$ $$-16 - 46 T + 8 T^{2} + T^{3}$$
$73$ $$8 - 12 T - 14 T^{2} + T^{3}$$
$79$ $$-32 - 32 T + 2 T^{2} + T^{3}$$
$83$ $$-232 - 62 T + 4 T^{2} + T^{3}$$
$89$ $$472 - 44 T - 10 T^{2} + T^{3}$$
$97$ $$-512 + 228 T - 28 T^{2} + T^{3}$$