# Properties

 Label 6384.2.a.ca Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.13068.1 Defining polynomial: $$x^{4} - x^{3} - 6 x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta_{2} ) q^{5} - q^{7} + q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -2 - \beta_{3} ) q^{13} + ( -1 - \beta_{2} ) q^{15} + ( -1 + \beta_{2} - \beta_{3} ) q^{17} + q^{19} - q^{21} + ( -\beta_{1} + 2 \beta_{2} ) q^{23} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} + q^{27} + ( 1 - \beta_{1} - \beta_{2} ) q^{29} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( \beta_{1} + \beta_{3} ) q^{33} + ( 1 + \beta_{2} ) q^{35} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -2 - \beta_{3} ) q^{39} -2 q^{41} + ( -2 + \beta_{1} + \beta_{3} ) q^{43} + ( -1 - \beta_{2} ) q^{45} + ( 1 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{47} + q^{49} + ( -1 + \beta_{2} - \beta_{3} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} ) q^{53} + ( -4 - 2 \beta_{3} ) q^{55} + q^{57} + ( -6 + 2 \beta_{1} ) q^{61} - q^{63} + ( 4 + 4 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{67} + ( -\beta_{1} + 2 \beta_{2} ) q^{69} + ( -1 - \beta_{1} - \beta_{2} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{75} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( -8 - \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{83} + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{85} + ( 1 - \beta_{1} - \beta_{2} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 2 + \beta_{3} ) q^{91} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{93} + ( -1 - \beta_{2} ) q^{95} + ( 2 - 3 \beta_{1} - \beta_{3} ) q^{97} + ( \beta_{1} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10})$$ $$4 q + 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 6 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{21} - 2 q^{23} + 8 q^{25} + 4 q^{27} + 2 q^{29} - 10 q^{31} + 4 q^{35} + 6 q^{37} - 6 q^{39} - 8 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 4 q^{57} - 20 q^{61} - 4 q^{63} + 12 q^{65} - 12 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} + 8 q^{75} - 34 q^{79} + 4 q^{81} - 6 q^{83} - 16 q^{85} + 2 q^{87} - 8 q^{89} + 6 q^{91} - 10 q^{93} - 4 q^{95} + 4 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} - x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 2 \nu^{2} - 12 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{2} + 4 \beta_{1} + 4$$

## 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
 −0.548230 0.328543 3.04374 −1.82405
0 1.00000 0 −4.42703 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.50541 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 −0.494592 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 2.42703 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.ca 4
4.b odd 2 1 3192.2.a.w 4
12.b even 2 1 9576.2.a.cl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3192.2.a.w 4 4.b odd 2 1
6384.2.a.ca 4 1.a even 1 1 trivial
9576.2.a.cl 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{4} + 4 T_{5}^{3} - 6 T_{5}^{2} - 20 T_{5} - 8$$ $$T_{11}^{4} - 36 T_{11}^{2} + 192$$ $$T_{13}^{4} + 6 T_{13}^{3} - 12 T_{13}^{2} - 96 T_{13} - 96$$ $$T_{17}^{4} + 2 T_{17}^{3} - 30 T_{17}^{2} - 64 T_{17} - 32$$ $$T_{23}^{4} + 2 T_{23}^{3} - 60 T_{23}^{2} - 160 T_{23} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$-8 - 20 T - 6 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$192 - 36 T^{2} + T^{4}$$
$13$ $$-96 - 96 T - 12 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$-32 - 64 T - 30 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$64 - 160 T - 60 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$-8 - 56 T - 42 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$-464 - 344 T - 24 T^{2} + 10 T^{3} + T^{4}$$
$37$ $$144 + 72 T - 48 T^{2} - 6 T^{3} + T^{4}$$
$41$ $$( 2 + T )^{4}$$
$43$ $$64 - 112 T - 12 T^{2} + 8 T^{3} + T^{4}$$
$47$ $$2776 + 212 T - 102 T^{2} - 4 T^{3} + T^{4}$$
$53$ $$-8 - 56 T - 42 T^{2} - 2 T^{3} + T^{4}$$
$59$ $$T^{4}$$
$61$ $$-3152 - 784 T + 48 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$-2976 - 1104 T - 60 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$-288 - 216 T - 30 T^{2} + 6 T^{3} + T^{4}$$
$73$ $$-1712 + 1136 T - 168 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$64 + 1312 T + 372 T^{2} + 34 T^{3} + T^{4}$$
$83$ $$3936 - 576 T - 150 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$2512 - 544 T - 120 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$-1856 + 1472 T - 204 T^{2} - 4 T^{3} + T^{4}$$