# Properties

 Label 8281.2.a.u Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $2$ CM discriminant -91 Inner twists $4$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{4} -\beta q^{5} -3 q^{9} +O(q^{10})$$ $$q -2 q^{4} -\beta q^{5} -3 q^{9} + 4 q^{16} -\beta q^{19} + 2 \beta q^{20} - q^{23} + 8 q^{25} -5 q^{29} + 3 \beta q^{31} + 6 q^{36} -2 \beta q^{41} -9 q^{43} + 3 \beta q^{45} + \beta q^{47} + 11 q^{53} + 4 \beta q^{59} -8 q^{64} -3 \beta q^{73} + 2 \beta q^{76} + 15 q^{79} -4 \beta q^{80} + 9 q^{81} + 5 \beta q^{83} + \beta q^{89} + 2 q^{92} + 13 q^{95} + 5 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 6 q^{9} + O(q^{10})$$ $$2 q - 4 q^{4} - 6 q^{9} + 8 q^{16} - 2 q^{23} + 16 q^{25} - 10 q^{29} + 12 q^{36} - 18 q^{43} + 22 q^{53} - 16 q^{64} + 30 q^{79} + 18 q^{81} + 4 q^{92} + 26 q^{95} + O(q^{100})$$

## 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
 2.30278 −1.30278
0 0 −2.00000 −3.60555 0 0 0 −3.00000 0
1.2 0 0 −2.00000 3.60555 0 0 0 −3.00000 0
 $$n$$: 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
91.b odd 2 1 CM by $$\Q(\sqrt{-91})$$
7.b odd 2 1 inner
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.u 2
7.b odd 2 1 inner 8281.2.a.u 2
13.b even 2 1 inner 8281.2.a.u 2
13.d odd 4 2 637.2.c.b 2
91.b odd 2 1 CM 8281.2.a.u 2
91.i even 4 2 637.2.c.b 2
91.z odd 12 4 637.2.r.b 4
91.bb even 12 4 637.2.r.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.b 2 13.d odd 4 2
637.2.c.b 2 91.i even 4 2
637.2.r.b 4 91.z odd 12 4
637.2.r.b 4 91.bb even 12 4
8281.2.a.u 2 1.a even 1 1 trivial
8281.2.a.u 2 7.b odd 2 1 inner
8281.2.a.u 2 13.b even 2 1 inner
8281.2.a.u 2 91.b odd 2 1 CM

## Hecke kernels

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

 $$T_{2}$$ $$T_{3}$$ $$T_{5}^{2} - 13$$ $$T_{11}$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-13 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$-13 + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$-117 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$-52 + T^{2}$$
$43$ $$( 9 + T )^{2}$$
$47$ $$-13 + T^{2}$$
$53$ $$( -11 + T )^{2}$$
$59$ $$-208 + T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-117 + T^{2}$$
$79$ $$( -15 + T )^{2}$$
$83$ $$-325 + T^{2}$$
$89$ $$-13 + T^{2}$$
$97$ $$-325 + T^{2}$$