# Properties

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

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 1.73205 1.73205 0 1.73205 −2.00000 −3.00000
1.2 1.73205 −1.00000 1.00000 −1.73205 −1.73205 0 −1.73205 −2.00000 −3.00000
 $$n$$: 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
13.b even 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.s 2
7.b odd 2 1 8281.2.a.w 2
7.c even 3 2 1183.2.e.e 4
13.b even 2 1 inner 8281.2.a.s 2
13.f odd 12 2 637.2.q.c 2
91.b odd 2 1 8281.2.a.w 2
91.r even 6 2 1183.2.e.e 4
91.w even 12 2 637.2.u.a 2
91.x odd 12 2 91.2.k.a 2
91.ba even 12 2 637.2.k.b 2
91.bc even 12 2 637.2.q.b 2
91.bd odd 12 2 91.2.u.a yes 2
273.bv even 12 2 819.2.bm.a 2
273.bw even 12 2 819.2.do.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 91.x odd 12 2
91.2.u.a yes 2 91.bd odd 12 2
637.2.k.b 2 91.ba even 12 2
637.2.q.b 2 91.bc even 12 2
637.2.q.c 2 13.f odd 12 2
637.2.u.a 2 91.w even 12 2
819.2.bm.a 2 273.bv even 12 2
819.2.do.c 2 273.bw even 12 2
1183.2.e.e 4 7.c even 3 2
1183.2.e.e 4 91.r even 6 2
8281.2.a.s 2 1.a even 1 1 trivial
8281.2.a.s 2 13.b even 2 1 inner
8281.2.a.w 2 7.b odd 2 1
8281.2.a.w 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{2} - 3$$ $$T_{3} + 1$$ $$T_{5}^{2} - 3$$ $$T_{11}^{2} - 27$$ $$T_{17} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-3 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-27 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$-3 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -3 + T )^{2}$$
$31$ $$-3 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-27 + T^{2}$$
$43$ $$( 11 + T )^{2}$$
$47$ $$-75 + T^{2}$$
$53$ $$( 9 + T )^{2}$$
$59$ $$-12 + T^{2}$$
$61$ $$( -7 + T )^{2}$$
$67$ $$-75 + T^{2}$$
$71$ $$-3 + T^{2}$$
$73$ $$-75 + T^{2}$$
$79$ $$( 5 + T )^{2}$$
$83$ $$-12 + T^{2}$$
$89$ $$-48 + T^{2}$$
$97$ $$-27 + T^{2}$$
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