Properties

 Label 6384.2.a.cf 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: 5.5.368464.1 Defining polynomial: $$x^{5} - 2 x^{4} - 6 x^{3} + 6 x^{2} + 6 x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 1 + \beta_{3} ) q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 1 + \beta_{3} ) q^{5} - q^{7} + q^{9} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{11} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( 1 + \beta_{3} ) q^{15} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{17} + q^{19} - q^{21} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( 3 + 2 \beta_{1} - \beta_{4} ) q^{25} + q^{27} + ( 1 - \beta_{2} + \beta_{4} ) q^{29} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{31} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{33} + ( -1 - \beta_{3} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{39} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{41} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( 1 + \beta_{3} ) q^{45} + ( 1 + \beta_{2} + \beta_{4} ) q^{47} + q^{49} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{51} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{53} + ( 1 + \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{55} + q^{57} -2 \beta_{4} q^{59} + ( -3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{61} - q^{63} + ( 5 + \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{65} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{69} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{71} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{73} + ( 3 + 2 \beta_{1} - \beta_{4} ) q^{75} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} ) q^{77} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{79} + q^{81} + ( -\beta_{1} + 3 \beta_{3} + 2 \beta_{4} ) q^{83} + ( 2 + 4 \beta_{3} - \beta_{4} ) q^{85} + ( 1 - \beta_{2} + \beta_{4} ) q^{87} + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{89} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{91} + ( 2 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{93} + ( 1 + \beta_{3} ) q^{95} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} + O(q^{10})$$ $$5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} + 2 q^{47} + 5 q^{49} + 12 q^{51} + 12 q^{55} + 5 q^{57} + 4 q^{59} - 14 q^{61} - 5 q^{63} + 32 q^{65} + 20 q^{67} - 12 q^{69} + 6 q^{71} + 10 q^{73} + 15 q^{75} + 8 q^{77} + 5 q^{81} - 6 q^{83} + 8 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} - 8 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 2 \nu + 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 7 \nu^{2} + 5$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} + 6 \nu^{2} - 4 \nu - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{4} + 3 \beta_{3} - \beta_{2} + 5 \beta_{1} + 13$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} + 26$$

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
 1.17837 −1.78948 −1.09027 3.14884 0.552543
0 1.00000 0 −3.42801 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.430991 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 0.388134 0 −1.00000 0 1.00000 0
1.4 0 1.00000 0 3.68348 0 −1.00000 0 1.00000 0
1.5 0 1.00000 0 3.78739 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.cf 5
4.b odd 2 1 399.2.a.g 5
12.b even 2 1 1197.2.a.o 5
20.d odd 2 1 9975.2.a.bp 5
28.d even 2 1 2793.2.a.bg 5
76.d even 2 1 7581.2.a.w 5
84.h odd 2 1 8379.2.a.cb 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.a.g 5 4.b odd 2 1
1197.2.a.o 5 12.b even 2 1
2793.2.a.bg 5 28.d even 2 1
6384.2.a.cf 5 1.a even 1 1 trivial
7581.2.a.w 5 76.d even 2 1
8379.2.a.cb 5 84.h odd 2 1
9975.2.a.bp 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6384))$$:

 $$T_{5}^{5} - 4 T_{5}^{4} - 12 T_{5}^{3} + 48 T_{5}^{2} + 4 T_{5} - 8$$ $$T_{11}^{5} + 8 T_{11}^{4} - 32 T_{11}^{3} - 304 T_{11}^{2} + 224 T_{11} + 2816$$ $$T_{13}^{5} + 6 T_{13}^{4} - 40 T_{13}^{3} - 224 T_{13}^{2} + 384 T_{13} + 1984$$ $$T_{17}^{5} - 12 T_{17}^{4} + 20 T_{17}^{3} + 248 T_{17}^{2} - 1116 T_{17} + 1256$$ $$T_{23}^{5} + 12 T_{23}^{4} - 16 T_{23}^{3} - 464 T_{23}^{2} - 352 T_{23} + 2432$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$( -1 + T )^{5}$$
$5$ $$-8 + 4 T + 48 T^{2} - 12 T^{3} - 4 T^{4} + T^{5}$$
$7$ $$( 1 + T )^{5}$$
$11$ $$2816 + 224 T - 304 T^{2} - 32 T^{3} + 8 T^{4} + T^{5}$$
$13$ $$1984 + 384 T - 224 T^{2} - 40 T^{3} + 6 T^{4} + T^{5}$$
$17$ $$1256 - 1116 T + 248 T^{2} + 20 T^{3} - 12 T^{4} + T^{5}$$
$19$ $$( -1 + T )^{5}$$
$23$ $$2432 - 352 T - 464 T^{2} - 16 T^{3} + 12 T^{4} + T^{5}$$
$29$ $$-8 - 4 T + 312 T^{2} - 64 T^{3} - 4 T^{4} + T^{5}$$
$31$ $$1408 - 2160 T + 944 T^{2} - 88 T^{3} - 8 T^{4} + T^{5}$$
$37$ $$-416 + 592 T + 48 T^{2} - 56 T^{3} - 2 T^{4} + T^{5}$$
$41$ $$-416 - 464 T + 768 T^{2} - 72 T^{3} - 10 T^{4} + T^{5}$$
$43$ $$1984 - 1616 T + 288 T^{2} + 48 T^{3} - 16 T^{4} + T^{5}$$
$47$ $$-32 + 412 T + 120 T^{2} - 56 T^{3} - 2 T^{4} + T^{5}$$
$53$ $$-5416 + 2444 T + 64 T^{2} - 104 T^{3} + T^{5}$$
$59$ $$-2048 + 1024 T + 256 T^{2} - 128 T^{3} - 4 T^{4} + T^{5}$$
$61$ $$16736 + 5200 T - 1360 T^{2} - 120 T^{3} + 14 T^{4} + T^{5}$$
$67$ $$-512 - 1088 T + 960 T^{2} + 16 T^{3} - 20 T^{4} + T^{5}$$
$71$ $$1696 + 900 T - 88 T^{2} - 92 T^{3} - 6 T^{4} + T^{5}$$
$73$ $$-11552 - 1520 T + 2448 T^{2} - 216 T^{3} - 10 T^{4} + T^{5}$$
$79$ $$-512 + 896 T - 192 T^{2} - 192 T^{3} + T^{5}$$
$83$ $$8912 + 3708 T - 560 T^{2} - 224 T^{3} + 6 T^{4} + T^{5}$$
$89$ $$-168352 - 16208 T + 5376 T^{2} - 72 T^{3} - 26 T^{4} + T^{5}$$
$97$ $$-4448 + 6480 T - 1520 T^{2} - 88 T^{3} + 18 T^{4} + T^{5}$$