# Properties

 Label 6384.2.a.cb Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.9248.1 Defining polynomial: $$x^{4} - 5 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{2} + 5 \beta_{1}$$$$)/2$$

## 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
 −2.13578 0.662153 −0.662153 2.13578
0 1.00000 0 −3.33513 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 −1.69614 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 1.69614 0 1.00000 0 1.00000 0
1.4 0 1.00000 0 3.33513 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.cb 4
4.b odd 2 1 3192.2.a.x 4
12.b even 2 1 9576.2.a.cj 4

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

 $$T_{5}^{4} - 14 T_{5}^{2} + 32$$ $$T_{11}^{4} - 20 T_{11}^{2} + 32$$ $$T_{13}$$ $$T_{17}^{4} + 4 T_{17}^{3} - 38 T_{17}^{2} - 152 T_{17} + 16$$ $$T_{23}^{4} + 4 T_{23}^{3} - 68 T_{23}^{2} - 416 T_{23} - 608$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$32 - 14 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$32 - 20 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$16 - 152 T - 38 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$-608 - 416 T - 68 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$104 + 24 T - 42 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$-32 + 48 T + 4 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$16 - 16 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$-16 - 128 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$-1664 + 608 T - 20 T^{2} - 12 T^{3} + T^{4}$$
$47$ $$2416 + 456 T - 98 T^{2} - 8 T^{3} + T^{4}$$
$53$ $$-472 + 248 T + 6 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$2048 - 112 T^{2} + T^{4}$$
$61$ $$208 + 288 T - 56 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$2144 + 864 T - 132 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$-416 + 304 T - 26 T^{2} - 12 T^{3} + T^{4}$$
$73$ $$-16 - 128 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-5312 + 1504 T + 4 T^{2} - 20 T^{3} + T^{4}$$
$83$ $$-1696 + 368 T + 70 T^{2} - 20 T^{3} + T^{4}$$
$89$ $$1328 - 544 T - 192 T^{2} + T^{4}$$
$97$ $$32 - 80 T - 4 T^{2} + 8 T^{3} + T^{4}$$