Properties

 Label 8016.2.a.u Level $8016$ Weight $2$ Character orbit 8016.a Self dual yes Analytic conductor $64.008$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8016 = 2^{4} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8016.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$64.0080822603$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.38569.1 Defining polynomial: $$x^{5} - 5 x^{3} + 4 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 501) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( \beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + q^{9} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{11} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{13} + ( \beta_{1} + \beta_{4} ) q^{15} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 4 - \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{21} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{23} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{25} + q^{27} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{29} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{33} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{35} + ( -4 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( -1 + \beta_{2} + 2 \beta_{4} ) q^{39} + ( 1 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{41} + ( 3 - \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{4} ) q^{45} + ( 1 + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{47} + ( -3 + 2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{49} + ( 1 + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 2 + \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} ) q^{53} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 4 - \beta_{2} - \beta_{3} ) q^{57} + ( 2 + 5 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{59} + ( -3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} ) q^{61} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( 5 - 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{65} + ( 3 + 6 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{67} + ( -3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{69} + ( -1 + 5 \beta_{1} + \beta_{2} + 8 \beta_{3} + 3 \beta_{4} ) q^{71} + ( 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{73} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{75} + ( 5 + \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{77} + ( 1 + 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{79} + q^{81} + ( 1 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} ) q^{83} + ( -2 + 5 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{85} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{87} + ( 4 \beta_{1} + \beta_{2} + 5 \beta_{3} + 4 \beta_{4} ) q^{89} + ( 2 + 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} ) q^{91} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{95} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{97} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + O(q^{10})$$ $$5q + 5q^{3} - q^{5} + 4q^{7} + 5q^{9} + 7q^{11} - 8q^{13} - q^{15} + 5q^{17} + 20q^{19} + 4q^{21} - q^{23} - 6q^{25} + 5q^{27} - 5q^{29} + 18q^{31} + 7q^{33} + 6q^{35} - 17q^{37} - 8q^{39} + 6q^{41} + 10q^{43} - q^{45} + 3q^{47} - 13q^{49} + 5q^{51} + 15q^{53} + 9q^{55} + 20q^{57} + 17q^{59} - 2q^{61} + 4q^{63} + 26q^{65} + 24q^{67} - q^{69} - q^{71} + 2q^{73} - 6q^{75} + 26q^{77} + 6q^{79} + 5q^{81} + 7q^{83} - 11q^{85} - 5q^{87} + 15q^{91} + 18q^{93} - q^{95} - 13q^{97} + 7q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 5 x^{3} + 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{4} - 4 \nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{4} + 5 \nu^{2} - 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + \nu^{3} - 5 \nu^{2} - 4 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{3} + 5 \beta_{2} + 7$$

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
 −1.95408 0.790734 −1.15098 0.275834 2.03850
0 1.00000 0 −3.11102 0 1.66149 0 1.00000 0
1.2 0 1.00000 0 −1.61314 0 −2.77861 0 1.00000 0
1.3 0 1.00000 0 0.0593478 0 1.53510 0 1.00000 0
1.4 0 1.00000 0 1.81885 0 0.619101 0 1.00000 0
1.5 0 1.00000 0 1.84596 0 2.96293 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$167$$ $$-1$$

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 8016.2.a.u 5
4.b odd 2 1 501.2.a.c 5
12.b even 2 1 1503.2.a.c 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.c 5 4.b odd 2 1
1503.2.a.c 5 12.b even 2 1
8016.2.a.u 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{5} + T_{5}^{4} - 9 T_{5}^{3} - 2 T_{5}^{2} + 17 T_{5} - 1$$ $$T_{7}^{5} - 4 T_{7}^{4} - 3 T_{7}^{3} + 29 T_{7}^{2} - 37 T_{7} + 13$$ $$T_{11}^{5} - 7 T_{11}^{4} - 3 T_{11}^{3} + 92 T_{11}^{2} - 101 T_{11} - 121$$ $$T_{13}^{5} + 8 T_{13}^{4} - T_{13}^{3} - 66 T_{13}^{2} - 48 T_{13} + 17$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$( -1 + T )^{5}$$
$5$ $$-1 + 17 T - 2 T^{2} - 9 T^{3} + T^{4} + T^{5}$$
$7$ $$13 - 37 T + 29 T^{2} - 3 T^{3} - 4 T^{4} + T^{5}$$
$11$ $$-121 - 101 T + 92 T^{2} - 3 T^{3} - 7 T^{4} + T^{5}$$
$13$ $$17 - 48 T - 66 T^{2} - T^{3} + 8 T^{4} + T^{5}$$
$17$ $$-293 - 255 T + 234 T^{2} - 34 T^{3} - 5 T^{4} + T^{5}$$
$19$ $$-599 + 940 T - 554 T^{2} + 153 T^{3} - 20 T^{4} + T^{5}$$
$23$ $$-1793 + 1301 T - 26 T^{2} - 77 T^{3} + T^{4} + T^{5}$$
$29$ $$4001 + 1033 T - 330 T^{2} - 71 T^{3} + 5 T^{4} + T^{5}$$
$31$ $$-181 + 444 T - 347 T^{2} + 118 T^{3} - 18 T^{4} + T^{5}$$
$37$ $$-13913 - 7275 T - 1084 T^{2} + 23 T^{3} + 17 T^{4} + T^{5}$$
$41$ $$1 - 14 T - 55 T^{2} - 40 T^{3} - 6 T^{4} + T^{5}$$
$43$ $$-4159 - 650 T + 689 T^{2} - 52 T^{3} - 10 T^{4} + T^{5}$$
$47$ $$-547 + 208 T + 223 T^{2} - 63 T^{3} - 3 T^{4} + T^{5}$$
$53$ $$707 - 4314 T + 1417 T^{2} - 65 T^{3} - 15 T^{4} + T^{5}$$
$59$ $$5737 - 10470 T + 2381 T^{2} - 73 T^{3} - 17 T^{4} + T^{5}$$
$61$ $$-8609 + 2152 T + 474 T^{2} - 147 T^{3} + 2 T^{4} + T^{5}$$
$67$ $$-23449 - 14296 T + 3132 T^{2} - 3 T^{3} - 24 T^{4} + T^{5}$$
$71$ $$28039 + 15837 T - 207 T^{2} - 292 T^{3} + T^{4} + T^{5}$$
$73$ $$8743 + 3298 T - 62 T^{2} - 109 T^{3} - 2 T^{4} + T^{5}$$
$79$ $$-31489 + 3112 T + 1471 T^{2} - 186 T^{3} - 6 T^{4} + T^{5}$$
$83$ $$7703 + 5178 T + 353 T^{2} - 129 T^{3} - 7 T^{4} + T^{5}$$
$89$ $$17071 + 5064 T - 322 T^{2} - 155 T^{3} + T^{5}$$
$97$ $$-869 - 1989 T - 1201 T^{2} - 112 T^{3} + 13 T^{4} + T^{5}$$