# Properties

 Label 8281.2.a.f Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 3q^{8} - 3q^{9} + O(q^{10})$$ $$q - q^{2} - q^{4} + 3q^{8} - 3q^{9} + 3q^{11} - q^{16} + 7q^{17} + 3q^{18} + 7q^{19} - 3q^{22} - 6q^{23} - 5q^{25} - 5q^{29} - 5q^{32} - 7q^{34} + 3q^{36} - 8q^{37} - 7q^{38} + 2q^{43} - 3q^{44} + 6q^{46} - 7q^{47} + 5q^{50} - 3q^{53} + 5q^{58} + 7q^{59} - 7q^{61} + 7q^{64} + 3q^{67} - 7q^{68} + 5q^{71} - 9q^{72} - 14q^{73} + 8q^{74} - 7q^{76} - 6q^{79} + 9q^{81} - 2q^{86} + 9q^{88} + 6q^{92} + 7q^{94} + 14q^{97} - 9q^{99} + O(q^{100})$$

## 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
 0
−1.00000 0 −1.00000 0 0 0 3.00000 −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

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.f 1
7.b odd 2 1 8281.2.a.e 1
7.c even 3 2 1183.2.e.b 2
13.b even 2 1 637.2.a.c 1
39.d odd 2 1 5733.2.a.c 1
91.b odd 2 1 637.2.a.d 1
91.r even 6 2 91.2.e.a 2
91.s odd 6 2 637.2.e.a 2
273.g even 2 1 5733.2.a.d 1
273.w odd 6 2 819.2.j.b 2
364.bl odd 6 2 1456.2.r.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.a 2 91.r even 6 2
637.2.a.c 1 13.b even 2 1
637.2.a.d 1 91.b odd 2 1
637.2.e.a 2 91.s odd 6 2
819.2.j.b 2 273.w odd 6 2
1183.2.e.b 2 7.c even 3 2
1456.2.r.g 2 364.bl odd 6 2
5733.2.a.c 1 39.d odd 2 1
5733.2.a.d 1 273.g even 2 1
8281.2.a.e 1 7.b odd 2 1
8281.2.a.f 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2} + 1$$ $$T_{3}$$ $$T_{5}$$ $$T_{11} - 3$$ $$T_{17} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$T$$
$17$ $$-7 + T$$
$19$ $$-7 + T$$
$23$ $$6 + T$$
$29$ $$5 + T$$
$31$ $$T$$
$37$ $$8 + T$$
$41$ $$T$$
$43$ $$-2 + T$$
$47$ $$7 + T$$
$53$ $$3 + T$$
$59$ $$-7 + T$$
$61$ $$7 + T$$
$67$ $$-3 + T$$
$71$ $$-5 + T$$
$73$ $$14 + T$$
$79$ $$6 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$-14 + T$$