# Properties

 Label 8281.2.a.bg Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# 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: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( -1 - \beta_{2} ) q^{8} + ( 3 - 2 \beta_{1} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{2} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 - \beta_{1} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( -1 - \beta_{2} ) q^{8} + ( 3 - 2 \beta_{1} ) q^{9} + ( 3 - \beta_{1} + \beta_{2} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + 4 q^{12} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{15} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{16} + ( -1 - \beta_{1} - \beta_{2} ) q^{17} + ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{18} + ( -1 - \beta_{1} ) q^{19} -2 \beta_{1} q^{20} + ( -2 + 2 \beta_{1} ) q^{22} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{23} -4 q^{24} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{25} + ( 4 - 4 \beta_{1} ) q^{27} + ( 8 + \beta_{2} ) q^{29} + ( 8 - 4 \beta_{1} + 2 \beta_{2} ) q^{30} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{32} + ( -6 + 2 \beta_{1} ) q^{33} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{34} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{36} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{37} + ( 3 + \beta_{1} + \beta_{2} ) q^{38} + 2 \beta_{1} q^{40} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{41} + ( 4 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} -4 q^{44} + ( 9 - 5 \beta_{1} + 2 \beta_{2} ) q^{45} + ( 7 - 3 \beta_{1} + 3 \beta_{2} ) q^{46} + ( -3 + \beta_{1} - 4 \beta_{2} ) q^{47} + ( -8 + 4 \beta_{1} ) q^{48} + ( 5 + \beta_{2} ) q^{50} + ( -2 - 2 \beta_{1} ) q^{51} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 12 - 4 \beta_{1} + 4 \beta_{2} ) q^{54} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} ) q^{57} + ( -1 - 9 \beta_{1} - \beta_{2} ) q^{58} + ( -2 + 2 \beta_{1} + 4 \beta_{2} ) q^{59} + ( 4 - 4 \beta_{1} ) q^{60} + 2 q^{61} + ( 1 - \beta_{1} - \beta_{2} ) q^{62} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{64} + ( -6 + 6 \beta_{1} - 2 \beta_{2} ) q^{66} + ( 2 + 6 \beta_{1} - 4 \beta_{2} ) q^{67} + ( -6 - 4 \beta_{1} - 2 \beta_{2} ) q^{68} + ( 5 - 9 \beta_{1} + 5 \beta_{2} ) q^{69} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{72} + ( -3 - \beta_{1} - 4 \beta_{2} ) q^{73} + ( 10 + 4 \beta_{2} ) q^{74} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{76} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 4 \beta_{1} - 2 \beta_{2} ) q^{80} + ( 3 - 6 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -4 + 2 \beta_{1} ) q^{82} + ( -1 - 9 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} ) q^{85} + ( 9 - \beta_{1} + 5 \beta_{2} ) q^{86} + ( 11 - 7 \beta_{1} + 7 \beta_{2} ) q^{87} + 4 q^{88} + ( -1 + 5 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 13 - 11 \beta_{1} + 3 \beta_{2} ) q^{90} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{92} + ( 7 + \beta_{1} - 3 \beta_{2} ) q^{93} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{94} + ( 2 + \beta_{2} ) q^{95} + ( -4 + 8 \beta_{1} - 4 \beta_{2} ) q^{96} + ( -3 - \beta_{1} ) q^{97} + ( -7 + 7 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{8} + 7q^{9} + O(q^{10})$$ $$3q - q^{2} + 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{8} + 7q^{9} + 8q^{10} - 2q^{11} + 12q^{12} + 6q^{15} - q^{16} - 4q^{17} + 15q^{18} - 4q^{19} - 2q^{20} - 4q^{22} + 10q^{23} - 12q^{24} - 5q^{25} + 8q^{27} + 24q^{29} + 20q^{30} - 4q^{31} - 7q^{32} - 16q^{33} + 14q^{34} - q^{36} + 10q^{38} + 2q^{40} + 2q^{41} + 10q^{43} - 12q^{44} + 22q^{45} + 18q^{46} - 8q^{47} - 20q^{48} + 15q^{50} - 8q^{51} + 8q^{53} + 32q^{54} - 6q^{55} + 2q^{57} - 12q^{58} - 4q^{59} + 8q^{60} + 6q^{61} + 2q^{62} - 17q^{64} - 12q^{66} + 12q^{67} - 22q^{68} + 6q^{69} + 6q^{71} + q^{72} - 10q^{73} + 30q^{74} + 16q^{75} - 8q^{76} - 14q^{79} - 14q^{80} + 3q^{81} - 10q^{82} - 12q^{83} + 10q^{85} + 26q^{86} + 26q^{87} + 12q^{88} + 2q^{89} + 28q^{90} - 12q^{92} + 22q^{93} + 10q^{94} + 6q^{95} - 4q^{96} - 10q^{97} - 14q^{99} + O(q^{100})$$

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

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

## 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
 2.34292 0.470683 −1.81361
−2.34292 1.14637 3.48929 −1.34292 −2.68585 0 −3.48929 −1.68585 3.14637
1.2 −0.470683 −2.24914 −1.77846 0.529317 1.05863 0 1.77846 2.05863 −0.249141
1.3 1.81361 3.10278 1.28917 2.81361 5.62721 0 −1.28917 6.62721 5.10278
 $$n$$: 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

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 8281.2.a.bg 3
7.b odd 2 1 1183.2.a.i 3
13.b even 2 1 637.2.a.j 3
39.d odd 2 1 5733.2.a.x 3
91.b odd 2 1 91.2.a.d 3
91.i even 4 2 1183.2.c.f 6
91.r even 6 2 637.2.e.i 6
91.s odd 6 2 637.2.e.j 6
273.g even 2 1 819.2.a.i 3
364.h even 2 1 1456.2.a.t 3
455.h odd 2 1 2275.2.a.m 3
728.b even 2 1 5824.2.a.bs 3
728.l odd 2 1 5824.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 91.b odd 2 1
637.2.a.j 3 13.b even 2 1
637.2.e.i 6 91.r even 6 2
637.2.e.j 6 91.s odd 6 2
819.2.a.i 3 273.g even 2 1
1183.2.a.i 3 7.b odd 2 1
1183.2.c.f 6 91.i even 4 2
1456.2.a.t 3 364.h even 2 1
2275.2.a.m 3 455.h odd 2 1
5733.2.a.x 3 39.d odd 2 1
5824.2.a.bs 3 728.b even 2 1
5824.2.a.by 3 728.l odd 2 1
8281.2.a.bg 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 2$$ $$T_{3}^{3} - 2 T_{3}^{2} - 6 T_{3} + 8$$ $$T_{5}^{3} - 2 T_{5}^{2} - 3 T_{5} + 2$$ $$T_{11}^{3} + 2 T_{11}^{2} - 6 T_{11} - 8$$ $$T_{17}^{3} + 4 T_{17}^{2} - 10 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 4 T + T^{2} + T^{3}$$
$3$ $$8 - 6 T - 2 T^{2} + T^{3}$$
$5$ $$2 - 3 T - 2 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$-8 - 6 T + 2 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$4 - 10 T + 4 T^{2} + T^{3}$$
$19$ $$-4 + T + 4 T^{2} + T^{3}$$
$23$ $$136 + T - 10 T^{2} + T^{3}$$
$29$ $$-454 + 185 T - 24 T^{2} + T^{3}$$
$31$ $$16 - 19 T + 4 T^{2} + T^{3}$$
$37$ $$124 - 58 T + T^{3}$$
$41$ $$-8 - 28 T - 2 T^{2} + T^{3}$$
$43$ $$628 - 71 T - 10 T^{2} + T^{3}$$
$47$ $$-544 - 79 T + 8 T^{2} + T^{3}$$
$53$ $$-22 - 35 T - 8 T^{2} + T^{3}$$
$59$ $$-688 - 156 T + 4 T^{2} + T^{3}$$
$61$ $$( -2 + T )^{3}$$
$67$ $$976 - 124 T - 12 T^{2} + T^{3}$$
$71$ $$-16 - 22 T - 6 T^{2} + T^{3}$$
$73$ $$-274 - 99 T + 10 T^{2} + T^{3}$$
$79$ $$-16 + 5 T + 14 T^{2} + T^{3}$$
$83$ $$-3268 - 271 T + 12 T^{2} + T^{3}$$
$89$ $$422 - 95 T - 2 T^{2} + T^{3}$$
$97$ $$22 + 29 T + 10 T^{2} + T^{3}$$