# Properties

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

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 \beta q^{4} + ( -1 - \beta ) q^{5} + ( 2 + 3 \beta ) q^{6} + ( 1 + 4 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 \beta q^{4} + ( -1 - \beta ) q^{5} + ( 2 + 3 \beta ) q^{6} + ( 1 + 4 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} + ( -2 - 3 \beta ) q^{10} + ( -3 + 3 \beta ) q^{11} + ( 3 + 6 \beta ) q^{12} + ( -2 - 3 \beta ) q^{15} + ( 5 + 3 \beta ) q^{16} + ( -5 + 4 \beta ) q^{17} + ( 2 + 5 \beta ) q^{18} + ( -3 + 3 \beta ) q^{19} + ( -3 - 6 \beta ) q^{20} + 3 \beta q^{22} + ( 2 - 4 \beta ) q^{23} + ( 5 + 9 \beta ) q^{24} + ( -3 + 3 \beta ) q^{25} + ( -1 + 2 \beta ) q^{27} + ( -1 + 5 \beta ) q^{29} + ( -5 - 8 \beta ) q^{30} + ( -5 + 6 \beta ) q^{31} + ( 6 + 3 \beta ) q^{32} + 3 \beta q^{33} + ( -1 + 3 \beta ) q^{34} + ( 9 + 6 \beta ) q^{36} + 4 q^{37} + 3 \beta q^{38} + ( -5 - 9 \beta ) q^{40} + ( -4 + 2 \beta ) q^{41} + ( -2 + 9 \beta ) q^{43} + 9 q^{44} + ( -2 - 5 \beta ) q^{45} + ( -2 - 6 \beta ) q^{46} + ( -1 + 2 \beta ) q^{47} + ( 8 + 11 \beta ) q^{48} + 3 \beta q^{50} + ( -1 + 3 \beta ) q^{51} + ( 7 - 2 \beta ) q^{53} + ( 1 + 3 \beta ) q^{54} -3 \beta q^{55} + 3 \beta q^{57} + ( 4 + 9 \beta ) q^{58} + ( -1 + 2 \beta ) q^{59} + ( -9 - 15 \beta ) q^{60} + 6 q^{61} + ( 1 + 7 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( 3 + 6 \beta ) q^{66} + ( -3 - 6 \beta ) q^{67} + ( 12 - 3 \beta ) q^{68} + ( -2 - 6 \beta ) q^{69} + ( 2 - 10 \beta ) q^{71} + ( 11 + 11 \beta ) q^{72} + 2 q^{73} + ( 4 + 4 \beta ) q^{74} + 3 \beta q^{75} + 9 q^{76} + 4 q^{79} + ( -8 - 11 \beta ) q^{80} + ( 4 - 6 \beta ) q^{81} -2 q^{82} + ( 3 - 6 \beta ) q^{83} + ( 1 - 3 \beta ) q^{85} + ( 7 + 16 \beta ) q^{86} + ( 4 + 9 \beta ) q^{87} + ( 9 + 3 \beta ) q^{88} + ( -13 + 5 \beta ) q^{89} + ( -7 - 12 \beta ) q^{90} + ( -12 - 6 \beta ) q^{92} + ( 1 + 7 \beta ) q^{93} + ( 1 + 3 \beta ) q^{94} -3 \beta q^{95} + ( 9 + 12 \beta ) q^{96} + ( -14 - 3 \beta ) q^{97} + ( 12 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 7q^{6} + 6q^{8} + q^{9} + O(q^{10})$$ $$2q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 7q^{6} + 6q^{8} + q^{9} - 7q^{10} - 3q^{11} + 12q^{12} - 7q^{15} + 13q^{16} - 6q^{17} + 9q^{18} - 3q^{19} - 12q^{20} + 3q^{22} + 19q^{24} - 3q^{25} + 3q^{29} - 18q^{30} - 4q^{31} + 15q^{32} + 3q^{33} + q^{34} + 24q^{36} + 8q^{37} + 3q^{38} - 19q^{40} - 6q^{41} + 5q^{43} + 18q^{44} - 9q^{45} - 10q^{46} + 27q^{48} + 3q^{50} + q^{51} + 12q^{53} + 5q^{54} - 3q^{55} + 3q^{57} + 17q^{58} - 33q^{60} + 12q^{61} + 9q^{62} + 4q^{64} + 12q^{66} - 12q^{67} + 21q^{68} - 10q^{69} - 6q^{71} + 33q^{72} + 4q^{73} + 12q^{74} + 3q^{75} + 18q^{76} + 8q^{79} - 27q^{80} + 2q^{81} - 4q^{82} - q^{85} + 30q^{86} + 17q^{87} + 21q^{88} - 21q^{89} - 26q^{90} - 30q^{92} + 9q^{93} + 5q^{94} - 3q^{95} + 30q^{96} - 31q^{97} + 21q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0.381966 0.381966 −1.85410 −0.381966 0.145898 0 −1.47214 −2.85410 −0.145898
1.2 2.61803 2.61803 4.85410 −2.61803 6.85410 0 7.47214 3.85410 −6.85410
 $$n$$: 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.bb 2
7.b odd 2 1 1183.2.a.g 2
13.b even 2 1 8281.2.a.n 2
13.c even 3 2 637.2.f.c 4
91.b odd 2 1 1183.2.a.c 2
91.g even 3 2 637.2.g.c 4
91.h even 3 2 637.2.h.f 4
91.i even 4 2 1183.2.c.c 4
91.m odd 6 2 637.2.g.b 4
91.n odd 6 2 91.2.f.a 4
91.v odd 6 2 637.2.h.g 4
273.bn even 6 2 819.2.o.c 4
364.v even 6 2 1456.2.s.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 91.n odd 6 2
637.2.f.c 4 13.c even 3 2
637.2.g.b 4 91.m odd 6 2
637.2.g.c 4 91.g even 3 2
637.2.h.f 4 91.h even 3 2
637.2.h.g 4 91.v odd 6 2
819.2.o.c 4 273.bn even 6 2
1183.2.a.c 2 91.b odd 2 1
1183.2.a.g 2 7.b odd 2 1
1183.2.c.c 4 91.i even 4 2
1456.2.s.h 4 364.v even 6 2
8281.2.a.n 2 13.b even 2 1
8281.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{2} - 3 T_{2} + 1$$ $$T_{3}^{2} - 3 T_{3} + 1$$ $$T_{5}^{2} + 3 T_{5} + 1$$ $$T_{11}^{2} + 3 T_{11} - 9$$ $$T_{17}^{2} + 6 T_{17} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + T^{2}$$
$3$ $$1 - 3 T + T^{2}$$
$5$ $$1 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-9 + 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-11 + 6 T + T^{2}$$
$19$ $$-9 + 3 T + T^{2}$$
$23$ $$-20 + T^{2}$$
$29$ $$-29 - 3 T + T^{2}$$
$31$ $$-41 + 4 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$4 + 6 T + T^{2}$$
$43$ $$-95 - 5 T + T^{2}$$
$47$ $$-5 + T^{2}$$
$53$ $$31 - 12 T + T^{2}$$
$59$ $$-5 + T^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-9 + 12 T + T^{2}$$
$71$ $$-116 + 6 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$-45 + T^{2}$$
$89$ $$79 + 21 T + T^{2}$$
$97$ $$229 + 31 T + T^{2}$$