# Properties

 Label 8281.2.a.bv Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{23})$$ Defining polynomial: $$x^{4} - 24 x^{2} + 121$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 24 x^{2} + 121$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 13 \nu$$$$)/11$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 12$$ $$\nu^{3}$$ $$=$$ $$11 \beta_{2} + 13 \beta_{1}$$

## 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
 2.68406 −4.09827 4.09827 −2.68406
1.00000 −1.41421 −1.00000 −4.09827 −1.41421 0 −3.00000 −1.00000 −4.09827
1.2 1.00000 −1.41421 −1.00000 2.68406 −1.41421 0 −3.00000 −1.00000 2.68406
1.3 1.00000 1.41421 −1.00000 −2.68406 1.41421 0 −3.00000 −1.00000 −2.68406
1.4 1.00000 1.41421 −1.00000 4.09827 1.41421 0 −3.00000 −1.00000 4.09827
 $$n$$: 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.bv 4
7.b odd 2 1 inner 8281.2.a.bv 4
13.b even 2 1 8281.2.a.bn 4
13.c even 3 2 637.2.f.g 8
91.b odd 2 1 8281.2.a.bn 4
91.g even 3 2 637.2.g.h 8
91.h even 3 2 637.2.h.k 8
91.m odd 6 2 637.2.g.h 8
91.n odd 6 2 637.2.f.g 8
91.v odd 6 2 637.2.h.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.g 8 13.c even 3 2
637.2.f.g 8 91.n odd 6 2
637.2.g.h 8 91.g even 3 2
637.2.g.h 8 91.m odd 6 2
637.2.h.k 8 91.h even 3 2
637.2.h.k 8 91.v odd 6 2
8281.2.a.bn 4 13.b even 2 1
8281.2.a.bn 4 91.b odd 2 1
8281.2.a.bv 4 1.a even 1 1 trivial
8281.2.a.bv 4 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{2} - 1$$ $$T_{3}^{2} - 2$$ $$T_{5}^{4} - 24 T_{5}^{2} + 121$$ $$T_{11}^{2} - 2 T_{11} - 22$$ $$T_{17}^{4} - 32 T_{17}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$121 - 24 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -22 - 2 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$49 - 32 T^{2} + T^{4}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$( -14 + 6 T + T^{2} )^{2}$$
$29$ $$( -7 + 8 T + T^{2} )^{2}$$
$31$ $$( -2 + T^{2} )^{2}$$
$37$ $$( -19 + 4 T + T^{2} )^{2}$$
$41$ $$841 - 104 T^{2} + T^{4}$$
$43$ $$( -14 - 6 T + T^{2} )^{2}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$( -83 + 6 T + T^{2} )^{2}$$
$59$ $$196 - 156 T^{2} + T^{4}$$
$61$ $$169 - 72 T^{2} + T^{4}$$
$67$ $$( -22 - 2 T + T^{2} )^{2}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$5329 - 192 T^{2} + T^{4}$$
$79$ $$( 26 + 14 T + T^{2} )^{2}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$33124 - 372 T^{2} + T^{4}$$
$97$ $$( -18 + T^{2} )^{2}$$