# Properties

 Label 8281.2.a.ba 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.

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

## 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 −2.23607 −1.85410 −2.23607 −0.854102 0 −1.47214 2.00000 −0.854102
1.2 2.61803 2.23607 4.85410 2.23607 5.85410 0 7.47214 2.00000 5.85410
 $$n$$: 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.ba 2
7.b odd 2 1 8281.2.a.z 2
7.d odd 6 2 1183.2.e.d 4
13.b even 2 1 637.2.a.e 2
39.d odd 2 1 5733.2.a.w 2
91.b odd 2 1 637.2.a.f 2
91.r even 6 2 637.2.e.h 4
91.s odd 6 2 91.2.e.b 4
273.g even 2 1 5733.2.a.v 2
273.ba even 6 2 819.2.j.c 4
364.x even 6 2 1456.2.r.j 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T + T^{2}$$
$3$ $$-5 + T^{2}$$
$5$ $$-5 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-11 - 6 T + T^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$31 + 12 T + T^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$-41 - 4 T + T^{2}$$
$41$ $$-20 + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-11 + 6 T + T^{2}$$
$53$ $$-11 + 6 T + T^{2}$$
$59$ $$-11 - 6 T + T^{2}$$
$61$ $$( 3 + T )^{2}$$
$67$ $$( -3 + T )^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$-29 - 8 T + T^{2}$$
$79$ $$-29 - 8 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$-5 + T^{2}$$
$97$ $$-164 + 8 T + T^{2}$$