# Properties

 Label 8281.2.a.ca Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.6995813.1 Defining polynomial: $$x^{6} - x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - x - 1$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{4} ) q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{5} ) q^{4} + \beta_{3} q^{5} + ( -2 + \beta_{2} - \beta_{5} ) q^{6} + ( -1 + \beta_{1} - \beta_{5} ) q^{8} + ( -\beta_{2} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{4} ) q^{2} -\beta_{4} q^{3} + ( 1 + \beta_{5} ) q^{4} + \beta_{3} q^{5} + ( -2 + \beta_{2} - \beta_{5} ) q^{6} + ( -1 + \beta_{1} - \beta_{5} ) q^{8} + ( -\beta_{2} + \beta_{5} ) q^{9} + ( 1 - \beta_{1} - \beta_{3} ) q^{10} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} ) q^{11} + ( 1 - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{12} -\beta_{5} q^{15} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{16} + ( 1 - \beta_{4} + \beta_{5} ) q^{17} + ( -1 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{18} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + ( \beta_{2} - \beta_{3} + \beta_{4} ) q^{20} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{22} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{23} + ( -2 + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{24} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{25} + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} ) q^{27} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{29} + ( 1 - \beta_{4} + \beta_{5} ) q^{30} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{31} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{32} + ( 3 - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 4 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{36} + ( 2 + 3 \beta_{3} + \beta_{5} ) q^{37} + ( -4 + 4 \beta_{1} + \beta_{3} - \beta_{5} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{40} + ( -2 - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{41} + ( 3 - 2 \beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{43} + ( -3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{44} + ( -1 - 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{45} + ( -4 + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{46} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{48} + ( -2 + 3 \beta_{1} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{50} + ( 4 - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{51} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{53} + ( -3 - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{54} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{55} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{57} + ( 1 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{58} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{59} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{60} + ( 2 - 6 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{62} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{64} + ( 5 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} ) q^{66} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{67} + ( 5 + \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{68} + ( 6 - \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{5} ) q^{69} + ( -2 - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{72} + ( -4 - 5 \beta_{1} + \beta_{4} ) q^{73} + ( 2 - 5 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{74} + ( -\beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{75} + ( -\beta_{1} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{76} + ( -2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{79} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{80} + ( 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{81} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{82} + ( 4 + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{83} + ( \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{85} + ( 2 - 3 \beta_{1} + \beta_{2} + 5 \beta_{4} - \beta_{5} ) q^{86} + ( 4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} ) q^{87} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{88} + ( 4 - 8 \beta_{1} + \beta_{2} + 5 \beta_{4} + 4 \beta_{5} ) q^{89} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{90} + ( 4 - 3 \beta_{2} + \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{92} + ( 1 + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{93} + ( 8 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{94} + ( -1 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{95} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} ) q^{96} + ( -8 + 2 \beta_{1} + \beta_{2} - 4 \beta_{4} - 3 \beta_{5} ) q^{97} + ( -3 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} + q^{3} + 4q^{4} + q^{5} - 9q^{6} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$6q - 2q^{2} + q^{3} + 4q^{4} + q^{5} - 9q^{6} - 3q^{8} - 3q^{9} + 4q^{10} - 4q^{11} + 5q^{12} + 2q^{15} - 8q^{16} + 5q^{17} - 3q^{18} - q^{19} - q^{20} + 5q^{22} + q^{23} - 11q^{24} - 7q^{25} + 4q^{27} - 3q^{29} + 5q^{30} + 16q^{31} - 8q^{32} + 16q^{33} - 16q^{34} + 21q^{36} + 13q^{37} - 17q^{38} - 5q^{40} - 8q^{41} + 11q^{43} - 21q^{44} - 7q^{45} - 16q^{46} - q^{47} + 21q^{48} - 6q^{50} + 20q^{51} + 2q^{53} - 18q^{54} + 9q^{55} + 21q^{57} + 8q^{58} + 13q^{59} - 20q^{60} - 5q^{61} + 5q^{62} - 15q^{64} + 18q^{66} + 11q^{67} + 29q^{68} + 23q^{69} - 6q^{71} - 25q^{72} - 30q^{73} + 3q^{74} - 3q^{75} - 9q^{76} - 7q^{79} - 7q^{80} + 6q^{81} + q^{82} + 27q^{83} + q^{85} + 7q^{86} + 16q^{87} + 4q^{89} + 8q^{90} + 27q^{92} + 7q^{93} + 45q^{94} + 6q^{95} + 19q^{96} - 35q^{97} - 10q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 6 x^{4} + 4 x^{3} + 7 x^{2} - x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} - \nu^{4} - 5 \nu^{3} + 4 \nu^{2} + 2 \nu - 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu^{4} - 6 \nu^{3} + 4 \nu^{2} + 7 \nu - 1$$ $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 2 \nu^{4} + 5 \nu^{3} - 9 \nu^{2} - 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{3} + 5 \beta_{2} + \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 5 \beta_{4} + 7 \beta_{3} + \beta_{2} + 24 \beta_{1} + 1$$

## 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
 −0.363441 2.33401 1.51235 −0.874884 −2.04394 0.435907
−2.38804 2.75148 3.70272 −0.982280 −6.57063 0 −4.06616 4.57063 2.34572
1.2 −1.90556 −0.428448 1.63116 1.47313 0.816433 0 0.702849 −2.81643 −2.80714
1.3 −0.851125 −0.661223 −1.27559 −3.44148 0.562784 0 2.78793 −2.56278 2.92913
1.4 −0.268125 1.14301 −1.92811 2.56175 −0.306470 0 1.05323 −1.69353 −0.686871
1.5 1.55469 0.489252 0.417051 1.19151 0.760633 0 −2.46099 −2.76063 1.85243
1.6 1.85816 −2.29407 1.45276 0.197362 −4.26275 0 −1.01686 2.26275 0.366731
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.ca 6
7.b odd 2 1 8281.2.a.bz 6
7.d odd 6 2 1183.2.e.h 12
13.b even 2 1 8281.2.a.cf 6
13.c even 3 2 637.2.f.j 12
91.b odd 2 1 8281.2.a.ce 6
91.g even 3 2 637.2.g.l 12
91.h even 3 2 637.2.h.l 12
91.m odd 6 2 91.2.g.b 12
91.n odd 6 2 637.2.f.k 12
91.s odd 6 2 1183.2.e.g 12
91.v odd 6 2 91.2.h.b yes 12
273.r even 6 2 819.2.s.d 12
273.bf even 6 2 819.2.n.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 91.m odd 6 2
91.2.h.b yes 12 91.v odd 6 2
637.2.f.j 12 13.c even 3 2
637.2.f.k 12 91.n odd 6 2
637.2.g.l 12 91.g even 3 2
637.2.h.l 12 91.h even 3 2
819.2.n.d 12 273.bf even 6 2
819.2.s.d 12 273.r even 6 2
1183.2.e.g 12 91.s odd 6 2
1183.2.e.h 12 7.d odd 6 2
8281.2.a.bz 6 7.b odd 2 1
8281.2.a.ca 6 1.a even 1 1 trivial
8281.2.a.ce 6 91.b odd 2 1
8281.2.a.cf 6 13.b even 2 1

## Hecke kernels

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

 $$T_{2}^{6} + 2 T_{2}^{5} - 6 T_{2}^{4} - 11 T_{2}^{3} + 8 T_{2}^{2} + 14 T_{2} + 3$$ $$T_{3}^{6} - T_{3}^{5} - 7 T_{3}^{4} + 4 T_{3}^{3} + 6 T_{3}^{2} - T_{3} - 1$$ $$T_{5}^{6} - T_{5}^{5} - 11 T_{5}^{4} + 18 T_{5}^{3} + 6 T_{5}^{2} - 17 T_{5} + 3$$ $$T_{11}^{6} + 4 T_{11}^{5} - 21 T_{11}^{4} - 76 T_{11}^{3} + 81 T_{11}^{2} + 207 T_{11} + 81$$ $$T_{17}^{6} - 5 T_{17}^{5} - 12 T_{17}^{4} + 14 T_{17}^{3} + 20 T_{17}^{2} - 8 T_{17} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 14 T + 8 T^{2} - 11 T^{3} - 6 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$-1 - T + 6 T^{2} + 4 T^{3} - 7 T^{4} - T^{5} + T^{6}$$
$5$ $$3 - 17 T + 6 T^{2} + 18 T^{3} - 11 T^{4} - T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$81 + 207 T + 81 T^{2} - 76 T^{3} - 21 T^{4} + 4 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$-9 - 8 T + 20 T^{2} + 14 T^{3} - 12 T^{4} - 5 T^{5} + T^{6}$$
$19$ $$873 + 1542 T + 629 T^{2} - 111 T^{3} - 64 T^{4} + T^{5} + T^{6}$$
$23$ $$-24387 - 440 T + 3031 T^{2} + 63 T^{3} - 106 T^{4} - T^{5} + T^{6}$$
$29$ $$-201 + 1124 T + 494 T^{2} - 244 T^{3} - 78 T^{4} + 3 T^{5} + T^{6}$$
$31$ $$-2477 + 4000 T - 2042 T^{2} + 295 T^{3} + 50 T^{4} - 16 T^{5} + T^{6}$$
$37$ $$-13477 + 17436 T - 7753 T^{2} + 1351 T^{3} - 38 T^{4} - 13 T^{5} + T^{6}$$
$41$ $$2043 + 1439 T - 283 T^{2} - 278 T^{3} - 21 T^{4} + 8 T^{5} + T^{6}$$
$43$ $$37 - 1620 T - 285 T^{2} + 266 T^{3} + T^{4} - 11 T^{5} + T^{6}$$
$47$ $$-17847 - 6323 T + 5684 T^{2} - 88 T^{3} - 177 T^{4} + T^{5} + T^{6}$$
$53$ $$-69 - 334 T + 1105 T^{2} + 186 T^{3} - 100 T^{4} - 2 T^{5} + T^{6}$$
$59$ $$9123 - 18461 T + 666 T^{2} + 996 T^{3} - 59 T^{4} - 13 T^{5} + T^{6}$$
$61$ $$32481 + 36801 T + 8972 T^{2} - 926 T^{3} - 201 T^{4} + 5 T^{5} + T^{6}$$
$67$ $$-16623 - 11067 T + 2270 T^{2} + 889 T^{3} - 106 T^{4} - 11 T^{5} + T^{6}$$
$71$ $$23043 + 13693 T - 103 T^{2} - 1136 T^{3} - 141 T^{4} + 6 T^{5} + T^{6}$$
$73$ $$-14029 - 24466 T - 8404 T^{2} - 251 T^{3} + 238 T^{4} + 30 T^{5} + T^{6}$$
$79$ $$10529 - 18957 T + 7249 T^{2} - 310 T^{3} - 148 T^{4} + 7 T^{5} + T^{6}$$
$83$ $$2673 - 1188 T - 1797 T^{2} + 403 T^{3} + 158 T^{4} - 27 T^{5} + T^{6}$$
$89$ $$-304479 + 79486 T + 32872 T^{2} + 132 T^{3} - 367 T^{4} - 4 T^{5} + T^{6}$$
$97$ $$-3899 - 8510 T - 1085 T^{2} + 1186 T^{3} + 365 T^{4} + 35 T^{5} + T^{6}$$