# Properties

 Label 8016.2.a.p 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: 5.5.36497.1 Defining polynomial: $$x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{9} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{13} + ( 2 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{15} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{17} + ( 3 + 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{23} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} - q^{27} + ( -1 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{29} + ( 4 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{33} + ( \beta_{1} + \beta_{2} - 4 \beta_{4} ) q^{35} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{39} + ( -4 + 6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{43} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{45} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{51} + ( 1 - 4 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -1 - 7 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{55} + ( -3 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{57} + ( 7 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( -1 - 6 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} ) q^{61} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + ( -6 + 5 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{65} + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{67} + ( -2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{69} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{71} + ( -1 - 2 \beta_{3} - \beta_{4} ) q^{73} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -9 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{77} + ( -3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{79} + q^{81} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - \beta_{4} ) q^{83} + ( 8 + 3 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{85} + ( 1 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{87} + ( -7 - 5 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} ) q^{89} + ( 6 + 6 \beta_{1} + \beta_{2} + 7 \beta_{4} ) q^{91} + ( -4 + 3 \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{93} + ( -9 - \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{95} + ( -2 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 5q^{3} - 9q^{5} + 4q^{7} + 5q^{9} + O(q^{10})$$ $$5q - 5q^{3} - 9q^{5} + 4q^{7} + 5q^{9} + 15q^{11} + 9q^{15} - 11q^{17} + 16q^{19} - 4q^{21} + 9q^{23} + 6q^{25} - 5q^{27} - q^{29} + 18q^{31} - 15q^{33} + 4q^{35} + 7q^{37} - 10q^{41} - 6q^{43} - 9q^{45} + 7q^{47} + 11q^{49} + 11q^{51} - 9q^{53} - 17q^{55} - 16q^{57} + 37q^{59} - 2q^{61} + 4q^{63} - 16q^{65} - 9q^{69} - 13q^{71} - 6q^{73} - 6q^{75} - 2q^{77} - 14q^{79} + 5q^{81} + 5q^{83} + 29q^{85} + q^{87} - 30q^{89} + 33q^{91} - 18q^{93} - 43q^{95} - 9q^{97} + 15q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1$$:

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

## 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
 2.31801 −1.55629 0.410375 1.33419 −0.506287
0 −1.00000 0 −3.48658 0 −3.13190 0 1.00000 0
1.2 0 −1.00000 0 −3.33576 0 0.969164 0 1.00000 0
1.3 0 −1.00000 0 −2.19483 0 4.52759 0 1.00000 0
1.4 0 −1.00000 0 −1.19539 0 −1.77240 0 1.00000 0
1.5 0 −1.00000 0 1.21255 0 3.40756 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.p 5
4.b odd 2 1 501.2.a.b 5
12.b even 2 1 1503.2.a.d 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.b 5 4.b odd 2 1
1503.2.a.d 5 12.b even 2 1
8016.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8016))$$:

 $$T_{5}^{5} + 9 T_{5}^{4} + 25 T_{5}^{3} + 12 T_{5}^{2} - 39 T_{5} - 37$$ $$T_{7}^{5} - 4 T_{7}^{4} - 15 T_{7}^{3} + 49 T_{7}^{2} + 55 T_{7} - 83$$ $$T_{11}^{5} - 15 T_{11}^{4} + 69 T_{11}^{3} - 36 T_{11}^{2} - 447 T_{11} + 761$$ $$T_{13}^{5} - 41 T_{13}^{3} - 32 T_{13}^{2} + 412 T_{13} + 687$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$-37 - 39 T + 12 T^{2} + 25 T^{3} + 9 T^{4} + T^{5}$$
$7$ $$-83 + 55 T + 49 T^{2} - 15 T^{3} - 4 T^{4} + T^{5}$$
$11$ $$761 - 447 T - 36 T^{2} + 69 T^{3} - 15 T^{4} + T^{5}$$
$13$ $$687 + 412 T - 32 T^{2} - 41 T^{3} + T^{5}$$
$17$ $$2203 - 343 T - 358 T^{2} - 10 T^{3} + 11 T^{4} + T^{5}$$
$19$ $$677 - 756 T + 102 T^{2} + 61 T^{3} - 16 T^{4} + T^{5}$$
$23$ $$47 - 355 T + 170 T^{2} - T^{3} - 9 T^{4} + T^{5}$$
$29$ $$-157 + 639 T - 146 T^{2} - 89 T^{3} + T^{4} + T^{5}$$
$31$ $$23 - 240 T - 79 T^{2} + 90 T^{3} - 18 T^{4} + T^{5}$$
$37$ $$1809 + 1645 T + 230 T^{2} - 77 T^{3} - 7 T^{4} + T^{5}$$
$41$ $$-87 + 656 T - 1119 T^{2} - 120 T^{3} + 10 T^{4} + T^{5}$$
$43$ $$127 - 1118 T - 953 T^{2} - 126 T^{3} + 6 T^{4} + T^{5}$$
$47$ $$-1209 - 668 T + 517 T^{2} - 57 T^{3} - 7 T^{4} + T^{5}$$
$53$ $$-3061 - 5578 T - 1725 T^{2} - 127 T^{3} + 9 T^{4} + T^{5}$$
$59$ $$617 + 3846 T - 2337 T^{2} + 469 T^{3} - 37 T^{4} + T^{5}$$
$61$ $$69523 + 13540 T - 854 T^{2} - 231 T^{3} + 2 T^{4} + T^{5}$$
$67$ $$-751 + 1608 T - 558 T^{2} - 219 T^{3} + T^{5}$$
$71$ $$64423 + 14869 T - 2677 T^{2} - 258 T^{3} + 13 T^{4} + T^{5}$$
$73$ $$-3 + 26 T - 40 T^{2} - 15 T^{3} + 6 T^{4} + T^{5}$$
$79$ $$-923 - 764 T - 119 T^{2} + 44 T^{3} + 14 T^{4} + T^{5}$$
$83$ $$10575 - 7840 T + 1991 T^{2} - 169 T^{3} - 5 T^{4} + T^{5}$$
$89$ $$-55203 - 18764 T - 1156 T^{2} + 217 T^{3} + 30 T^{4} + T^{5}$$
$97$ $$3391 + 555 T - 429 T^{2} - 48 T^{3} + 9 T^{4} + T^{5}$$