# Properties

 Label 8281.2.a.bo Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.105456.1 Defining polynomial: $$x^{4} - 13 x^{2} + 39$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{1} q^{5} + \beta_{3} q^{6} -3 q^{8} + ( 4 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{4} -\beta_{1} q^{5} + \beta_{3} q^{6} -3 q^{8} + ( 4 + \beta_{2} ) q^{9} -\beta_{3} q^{10} + ( 2 + 3 \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{12} + ( -7 - \beta_{2} ) q^{15} + ( -2 - \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 3 + 3 \beta_{2} ) q^{18} -\beta_{1} q^{19} + ( -\beta_{1} + \beta_{3} ) q^{20} + ( 9 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} -3 \beta_{1} q^{24} + ( 2 + \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{3} ) q^{27} + \beta_{2} q^{29} + ( -3 - 6 \beta_{2} ) q^{30} + ( \beta_{1} - \beta_{3} ) q^{31} + ( 3 - \beta_{2} ) q^{32} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{34} + ( 1 - 2 \beta_{2} ) q^{36} + ( -4 + 2 \beta_{2} ) q^{37} -\beta_{3} q^{38} + 3 \beta_{1} q^{40} -2 \beta_{3} q^{41} + ( -1 + 5 \beta_{2} ) q^{43} + ( -7 + 4 \beta_{2} ) q^{44} + ( -4 \beta_{1} - \beta_{3} ) q^{45} + ( 6 + 2 \beta_{2} ) q^{46} + ( 3 \beta_{1} + \beta_{3} ) q^{47} + ( -2 \beta_{1} - \beta_{3} ) q^{48} + ( 3 + \beta_{2} ) q^{50} + ( 4 - 5 \beta_{2} ) q^{51} + ( 7 + 2 \beta_{2} ) q^{53} + 3 \beta_{1} q^{54} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{55} + ( -7 - \beta_{2} ) q^{57} + ( 3 - \beta_{2} ) q^{58} + ( -\beta_{1} - \beta_{3} ) q^{59} + ( -4 + 5 \beta_{2} ) q^{60} + 2 \beta_{1} q^{61} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{62} + ( 1 + 6 \beta_{2} ) q^{64} + ( 9 \beta_{1} - \beta_{3} ) q^{66} - q^{67} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{68} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{69} -4 q^{71} + ( -12 - 3 \beta_{2} ) q^{72} + 2 \beta_{1} q^{73} + ( 6 - 6 \beta_{2} ) q^{74} + ( 2 \beta_{1} + \beta_{3} ) q^{75} + ( -\beta_{1} + \beta_{3} ) q^{76} + ( -2 + 2 \beta_{2} ) q^{79} + ( 2 \beta_{1} + \beta_{3} ) q^{80} + ( -2 + 4 \beta_{2} ) q^{81} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{82} + ( \beta_{1} + \beta_{3} ) q^{83} + ( -4 + 5 \beta_{2} ) q^{85} + ( 15 - 6 \beta_{2} ) q^{86} + \beta_{3} q^{87} + ( -6 - 9 \beta_{2} ) q^{88} -3 \beta_{1} q^{89} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{90} -2 q^{92} + ( 4 - 5 \beta_{2} ) q^{93} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{94} + ( 7 + \beta_{2} ) q^{95} + ( 3 \beta_{1} - \beta_{3} ) q^{96} + ( 4 \beta_{1} - \beta_{3} ) q^{97} + ( 17 + 11 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + O(q^{10})$$ $$4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46} + 10 q^{50} + 26 q^{51} + 24 q^{53} - 26 q^{57} + 14 q^{58} - 26 q^{60} - 8 q^{64} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 36 q^{74} - 12 q^{79} - 16 q^{81} - 26 q^{85} + 72 q^{86} - 6 q^{88} - 8 q^{92} + 26 q^{93} + 26 q^{95} + 46 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 13 x^{2} + 39$$:

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

## 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
 −2.16731 2.16731 −2.88145 2.88145
−2.30278 −2.16731 3.30278 2.16731 4.99082 0 −3.00000 1.69722 −4.99082
1.2 −2.30278 2.16731 3.30278 −2.16731 −4.99082 0 −3.00000 1.69722 4.99082
1.3 1.30278 −2.88145 −0.302776 2.88145 −3.75389 0 −3.00000 5.30278 3.75389
1.4 1.30278 2.88145 −0.302776 −2.88145 3.75389 0 −3.00000 5.30278 −3.75389
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.bo 4
7.b odd 2 1 inner 8281.2.a.bo 4
13.b even 2 1 8281.2.a.bu 4
13.e even 6 2 637.2.f.h 8
91.b odd 2 1 8281.2.a.bu 4
91.k even 6 2 637.2.h.j 8
91.l odd 6 2 637.2.h.j 8
91.p odd 6 2 637.2.g.i 8
91.t odd 6 2 637.2.f.h 8
91.u even 6 2 637.2.g.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h 8 13.e even 6 2
637.2.f.h 8 91.t odd 6 2
637.2.g.i 8 91.p odd 6 2
637.2.g.i 8 91.u even 6 2
637.2.h.j 8 91.k even 6 2
637.2.h.j 8 91.l odd 6 2
8281.2.a.bo 4 1.a even 1 1 trivial
8281.2.a.bo 4 7.b odd 2 1 inner
8281.2.a.bu 4 13.b even 2 1
8281.2.a.bu 4 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{2} + T_{2} - 3$$ $$T_{3}^{4} - 13 T_{3}^{2} + 39$$ $$T_{5}^{4} - 13 T_{5}^{2} + 39$$ $$T_{11}^{2} - T_{11} - 29$$ $$T_{17}^{4} - 52 T_{17}^{2} + 39$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -3 + T + T^{2} )^{2}$$
$3$ $$39 - 13 T^{2} + T^{4}$$
$5$ $$39 - 13 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -29 - T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$39 - 52 T^{2} + T^{4}$$
$19$ $$39 - 13 T^{2} + T^{4}$$
$23$ $$( -4 - 6 T + T^{2} )^{2}$$
$29$ $$( -3 + T + T^{2} )^{2}$$
$31$ $$39 - 52 T^{2} + T^{4}$$
$37$ $$( 12 + 10 T + T^{2} )^{2}$$
$41$ $$5616 - 156 T^{2} + T^{4}$$
$43$ $$( -69 + 7 T + T^{2} )^{2}$$
$47$ $$351 - 156 T^{2} + T^{4}$$
$53$ $$( 23 - 12 T + T^{2} )^{2}$$
$59$ $$351 - 52 T^{2} + T^{4}$$
$61$ $$624 - 52 T^{2} + T^{4}$$
$67$ $$( 1 + T )^{4}$$
$71$ $$( 4 + T )^{4}$$
$73$ $$624 - 52 T^{2} + T^{4}$$
$79$ $$( -4 + 6 T + T^{2} )^{2}$$
$83$ $$351 - 52 T^{2} + T^{4}$$
$89$ $$3159 - 117 T^{2} + T^{4}$$
$97$ $$11271 - 247 T^{2} + T^{4}$$