# Properties

 Label 6384.2.a.cc Level $6384$ Weight $2$ Character orbit 6384.a Self dual yes Analytic conductor $50.976$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.1240016.1 Defining polynomial: $$x^{5} - x^{4} - 8 x^{3} + 2 x^{2} + 16 x + 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 399) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9} + \beta_{1} q^{11} + ( 2 - \beta_{3} ) q^{13} -\beta_{2} q^{15} + ( -1 - \beta_{2} - \beta_{4} ) q^{17} - q^{19} + q^{21} -\beta_{1} q^{23} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} + ( \beta_{1} + \beta_{2} ) q^{29} + ( -2 + \beta_{1} ) q^{31} -\beta_{1} q^{33} -\beta_{2} q^{35} + ( 1 + 2 \beta_{2} - \beta_{4} ) q^{37} + ( -2 + \beta_{3} ) q^{39} + ( 1 - \beta_{3} + \beta_{4} ) q^{41} + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{43} + \beta_{2} q^{45} + ( 6 - \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( 1 + \beta_{2} + \beta_{4} ) q^{51} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{53} + ( 1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{55} + q^{57} + 2 \beta_{1} q^{59} + ( 3 + 2 \beta_{2} + \beta_{4} ) q^{61} - q^{63} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{65} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{67} + \beta_{1} q^{69} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{71} + ( 1 + 2 \beta_{3} - \beta_{4} ) q^{73} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{75} -\beta_{1} q^{77} + ( -2 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{79} + q^{81} + ( 5 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{83} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -\beta_{1} - \beta_{2} ) q^{87} + ( 1 - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{89} + ( -2 + \beta_{3} ) q^{91} + ( 2 - \beta_{1} ) q^{93} -\beta_{2} q^{95} + ( -5 + 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{97} + \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + O(q^{10})$$ $$5 q - 5 q^{3} - 2 q^{5} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} + 5 q^{21} - 2 q^{23} + 11 q^{25} - 5 q^{27} - 8 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} - 20 q^{43} - 2 q^{45} + 26 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 4 q^{55} + 5 q^{57} + 4 q^{59} + 10 q^{61} - 5 q^{63} - 4 q^{65} - 10 q^{67} + 2 q^{69} - 10 q^{71} + 10 q^{73} - 11 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 34 q^{83} - 36 q^{85} + 10 q^{89} - 8 q^{91} + 8 q^{93} + 2 q^{95} - 16 q^{97} + 2 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 8 x^{3} + 2 x^{2} + 16 x + 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 6$$ $$\beta_{4}$$ $$=$$ $$2 \nu^{4} - 4 \nu^{3} - 10 \nu^{2} + 12 \nu + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 6$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{4} + 7 \beta_{3} + 4 \beta_{2} + 9 \beta_{1} + 31$$$$)/2$$

## 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
 1.91889 −1.89281 −1.36162 −0.437507 2.77304
0 −1.00000 0 −4.29208 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.79287 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 1.06804 0 −1.00000 0 1.00000 0
1.4 0 −1.00000 0 1.47487 0 −1.00000 0 1.00000 0
1.5 0 −1.00000 0 2.54204 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.cc 5
4.b odd 2 1 399.2.a.f 5
12.b even 2 1 1197.2.a.p 5
20.d odd 2 1 9975.2.a.bq 5
28.d even 2 1 2793.2.a.be 5
76.d even 2 1 7581.2.a.x 5
84.h odd 2 1 8379.2.a.ce 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.a.f 5 4.b odd 2 1
1197.2.a.p 5 12.b even 2 1
2793.2.a.be 5 28.d even 2 1
6384.2.a.cc 5 1.a even 1 1 trivial
7581.2.a.x 5 76.d even 2 1
8379.2.a.ce 5 84.h odd 2 1
9975.2.a.bq 5 20.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{5} + 2 T_{5}^{4} - 16 T_{5}^{3} - 8 T_{5}^{2} + 68 T_{5} - 48$$ $$T_{11}^{5} - 2 T_{11}^{4} - 32 T_{11}^{3} + 16 T_{11}^{2} + 256 T_{11} + 192$$ $$T_{13}^{5} - 8 T_{13}^{4} - 8 T_{13}^{3} + 112 T_{13}^{2} + 32 T_{13} - 256$$ $$T_{17}^{5} + 2 T_{17}^{4} - 72 T_{17}^{3} - 176 T_{17}^{2} + 1108 T_{17} + 3168$$ $$T_{23}^{5} + 2 T_{23}^{4} - 32 T_{23}^{3} - 16 T_{23}^{2} + 256 T_{23} - 192$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$-48 + 68 T - 8 T^{2} - 16 T^{3} + 2 T^{4} + T^{5}$$
$7$ $$( 1 + T )^{5}$$
$11$ $$192 + 256 T + 16 T^{2} - 32 T^{3} - 2 T^{4} + T^{5}$$
$13$ $$-256 + 32 T + 112 T^{2} - 8 T^{3} - 8 T^{4} + T^{5}$$
$17$ $$3168 + 1108 T - 176 T^{2} - 72 T^{3} + 2 T^{4} + T^{5}$$
$19$ $$( 1 + T )^{5}$$
$23$ $$-192 + 256 T - 16 T^{2} - 32 T^{3} + 2 T^{4} + T^{5}$$
$29$ $$24 + 28 T - 80 T^{2} - 56 T^{3} + T^{5}$$
$31$ $$512 - 48 T - 144 T^{2} - 8 T^{3} + 8 T^{4} + T^{5}$$
$37$ $$608 + 2128 T + 176 T^{2} - 120 T^{3} - 2 T^{4} + T^{5}$$
$41$ $$96 + 176 T - 96 T^{2} - 72 T^{3} - 2 T^{4} + T^{5}$$
$43$ $$-13184 - 8272 T - 1376 T^{2} + 32 T^{3} + 20 T^{4} + T^{5}$$
$47$ $$3648 - 836 T - 592 T^{2} + 224 T^{3} - 26 T^{4} + T^{5}$$
$53$ $$20376 + 8812 T + 248 T^{2} - 192 T^{3} - 4 T^{4} + T^{5}$$
$59$ $$6144 + 4096 T + 128 T^{2} - 128 T^{3} - 4 T^{4} + T^{5}$$
$61$ $$-3872 - 304 T + 944 T^{2} - 88 T^{3} - 10 T^{4} + T^{5}$$
$67$ $$40064 + 3520 T - 1408 T^{2} - 136 T^{3} + 10 T^{4} + T^{5}$$
$71$ $$3888 - 236 T - 896 T^{2} - 84 T^{3} + 10 T^{4} + T^{5}$$
$73$ $$-32 - 5232 T + 1744 T^{2} - 120 T^{3} - 10 T^{4} + T^{5}$$
$79$ $$4864 + 1024 T - 512 T^{2} - 72 T^{3} + 14 T^{4} + T^{5}$$
$83$ $$159888 - 45476 T + 2872 T^{2} + 248 T^{3} - 34 T^{4} + T^{5}$$
$89$ $$-114336 + 24176 T + 2304 T^{2} - 312 T^{3} - 10 T^{4} + T^{5}$$
$97$ $$194816 + 19536 T - 4544 T^{2} - 312 T^{3} + 16 T^{4} + T^{5}$$