# Properties

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

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

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

## 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
 −1.41421 1.41421
−1.41421 1.41421 0 4.41421 −2.00000 0 2.82843 −1.00000 −6.24264
1.2 1.41421 −1.41421 0 1.58579 −2.00000 0 −2.82843 −1.00000 2.24264
 $$n$$: 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.v 2
7.b odd 2 1 1183.2.a.d 2
13.b even 2 1 637.2.a.g 2
39.d odd 2 1 5733.2.a.s 2
91.b odd 2 1 91.2.a.c 2
91.i even 4 2 1183.2.c.d 4
91.r even 6 2 637.2.e.g 4
91.s odd 6 2 637.2.e.f 4
273.g even 2 1 819.2.a.h 2
364.h even 2 1 1456.2.a.q 2
455.h odd 2 1 2275.2.a.j 2
728.b even 2 1 5824.2.a.bk 2
728.l odd 2 1 5824.2.a.bl 2

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

 $$T_{2}^{2} - 2$$ $$T_{3}^{2} - 2$$ $$T_{5}^{2} - 6 T_{5} + 7$$ $$T_{11}^{2} - 18$$ $$T_{17}^{2} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$7 - 6 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-18 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-9 + 6 T + T^{2}$$
$23$ $$1 + 6 T + T^{2}$$
$29$ $$1 - 6 T + T^{2}$$
$31$ $$-17 + 2 T + T^{2}$$
$37$ $$-14 - 4 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$7 - 6 T + T^{2}$$
$53$ $$1 + 6 T + T^{2}$$
$59$ $$4 - 12 T + T^{2}$$
$61$ $$( 6 + T )^{2}$$
$67$ $$-36 - 12 T + T^{2}$$
$71$ $$-14 - 12 T + T^{2}$$
$73$ $$7 + 10 T + T^{2}$$
$79$ $$-23 - 14 T + T^{2}$$
$83$ $$63 - 18 T + T^{2}$$
$89$ $$7 - 6 T + T^{2}$$
$97$ $$-161 + 2 T + T^{2}$$
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