# Properties

 Label 8281.2.a.ci Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $8$ CM no Inner twists $2$

# 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: $$8$$ Coefficient field: 8.8.8446345216.1 Defining polynomial: $$x^{8} - 8 x^{6} + 19 x^{4} - 14 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + ( 1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{3} - \beta_{7} ) q^{5} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 - \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{5} ) q^{2} + ( -\beta_{1} - \beta_{4} ) q^{3} + ( 1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{3} - \beta_{7} ) q^{5} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 - \beta_{5} + \beta_{6} ) q^{8} + ( 1 + \beta_{5} ) q^{9} + ( \beta_{1} - 2 \beta_{3} + 2 \beta_{7} ) q^{10} + ( 1 + \beta_{5} ) q^{11} + ( 2 \beta_{3} - \beta_{4} - 2 \beta_{7} ) q^{12} + ( 1 - \beta_{2} - 2 \beta_{6} ) q^{15} + ( \beta_{2} - \beta_{5} - \beta_{6} ) q^{16} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{17} + ( -3 + \beta_{2} + \beta_{5} + \beta_{6} ) q^{18} + ( -\beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{7} ) q^{20} + ( -3 + \beta_{2} + \beta_{5} + \beta_{6} ) q^{22} + ( -1 + \beta_{2} + \beta_{5} + 2 \beta_{6} ) q^{23} + ( -3 \beta_{3} - 2 \beta_{4} + 3 \beta_{7} ) q^{24} + ( -1 + \beta_{5} - 2 \beta_{6} ) q^{25} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{27} + ( -2 - 3 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{29} + ( -4 + \beta_{2} - \beta_{5} + 4 \beta_{6} ) q^{30} + ( -\beta_{4} + 3 \beta_{7} ) q^{31} + ( -2 \beta_{2} - \beta_{6} ) q^{32} + ( -2 \beta_{1} + \beta_{3} - \beta_{4} ) q^{33} + ( -4 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} ) q^{34} + ( 1 + 3 \beta_{5} - \beta_{6} ) q^{36} + ( 3 + 3 \beta_{2} + 4 \beta_{5} + \beta_{6} ) q^{37} + ( -6 \beta_{1} + 5 \beta_{3} + \beta_{4} - \beta_{7} ) q^{38} + ( 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{41} + ( -3 + 5 \beta_{2} + 2 \beta_{5} ) q^{43} + ( 1 + 3 \beta_{5} - \beta_{6} ) q^{44} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{45} + ( 2 + 3 \beta_{5} - 3 \beta_{6} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{47} + ( \beta_{1} + 2 \beta_{4} - 4 \beta_{7} ) q^{48} + ( -3 + \beta_{2} + 3 \beta_{5} + 5 \beta_{6} ) q^{50} + ( -6 + 3 \beta_{2} - \beta_{5} + \beta_{6} ) q^{51} + ( -1 - 4 \beta_{2} - \beta_{5} + \beta_{6} ) q^{53} + ( -4 \beta_{1} + \beta_{3} + 2 \beta_{7} ) q^{54} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{7} ) q^{55} + ( -6 + 2 \beta_{2} + 3 \beta_{6} ) q^{57} + ( 5 + \beta_{2} - 2 \beta_{5} - 6 \beta_{6} ) q^{58} + ( -3 \beta_{1} + \beta_{3} - 4 \beta_{4} + 2 \beta_{7} ) q^{59} + ( 9 + 2 \beta_{5} - 5 \beta_{6} ) q^{60} + ( \beta_{1} + 3 \beta_{3} + \beta_{4} - 2 \beta_{7} ) q^{61} + ( -\beta_{1} + 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{7} ) q^{62} + ( -3 + 2 \beta_{5} + 4 \beta_{6} ) q^{64} + ( 2 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{7} ) q^{66} + ( -2 + \beta_{2} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( 2 \beta_{1} - 3 \beta_{3} + 3 \beta_{4} + 5 \beta_{7} ) q^{68} + ( -4 \beta_{3} + 5 \beta_{7} ) q^{69} + ( -3 - 5 \beta_{2} - 4 \beta_{5} + \beta_{6} ) q^{71} + ( -2 + \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{72} + ( -\beta_{1} - 3 \beta_{3} - 7 \beta_{4} + 4 \beta_{7} ) q^{73} + ( -7 + \beta_{2} + 5 \beta_{5} + 2 \beta_{6} ) q^{74} + ( 5 \beta_{3} + 3 \beta_{4} - 6 \beta_{7} ) q^{75} + ( -\beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 9 \beta_{7} ) q^{76} + ( -1 + 7 \beta_{2} - \beta_{5} ) q^{79} + ( \beta_{3} - \beta_{4} ) q^{80} + ( -9 - \beta_{2} - 4 \beta_{5} - \beta_{6} ) q^{81} + ( 6 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} - 4 \beta_{7} ) q^{82} + ( -\beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{7} ) q^{83} + ( -5 - \beta_{5} + 4 \beta_{6} ) q^{85} + ( 4 - 3 \beta_{2} + 7 \beta_{5} + 2 \beta_{6} ) q^{86} + ( 4 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 9 \beta_{7} ) q^{87} + ( -2 + \beta_{2} + 3 \beta_{5} + 3 \beta_{6} ) q^{88} + ( 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 10 \beta_{7} ) q^{89} + ( 2 \beta_{1} - 4 \beta_{3} + \beta_{4} + 5 \beta_{7} ) q^{90} + ( -9 + \beta_{2} + 2 \beta_{5} + 5 \beta_{6} ) q^{92} + ( -1 - 4 \beta_{2} + 3 \beta_{6} ) q^{93} + ( -4 \beta_{1} + 5 \beta_{3} + 4 \beta_{4} + \beta_{7} ) q^{94} + ( -7 + \beta_{2} - \beta_{5} + 5 \beta_{6} ) q^{95} + ( 4 \beta_{3} - \beta_{4} - \beta_{7} ) q^{96} + ( 6 \beta_{1} + \beta_{3} - \beta_{4} ) q^{97} + ( 3 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} + 12q^{4} + 12q^{8} + 4q^{9} + O(q^{10})$$ $$8q - 4q^{2} + 12q^{4} + 12q^{8} + 4q^{9} + 4q^{11} + 8q^{15} + 4q^{16} - 28q^{18} - 28q^{22} - 12q^{23} - 12q^{25} - 8q^{29} - 28q^{30} - 4q^{36} + 8q^{37} - 32q^{43} - 4q^{44} + 4q^{46} - 36q^{50} - 44q^{51} - 4q^{53} - 48q^{57} + 48q^{58} + 64q^{60} - 32q^{64} - 20q^{67} - 8q^{71} - 28q^{72} - 76q^{74} - 4q^{79} - 56q^{81} - 36q^{85} + 4q^{86} - 28q^{88} - 80q^{92} - 8q^{93} - 52q^{95} + 28q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 19 x^{4} - 14 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - 6 \nu^{3} + 7 \nu$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} - 6 \nu^{4} + 7 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$-\nu^{6} + 7 \nu^{4} - 12 \nu^{2} + 3$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} - 7 \nu^{5} + 13 \nu^{3} - 6 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} + 5 \beta_{2} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{4} + 6 \beta_{3} + 11 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$6 \beta_{6} + 7 \beta_{5} + 23 \beta_{2} + 28$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} + 7 \beta_{4} + 29 \beta_{3} + 44 \beta_{1}$$

## 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
 1.11758 −1.11758 0.282452 −0.282452 2.09282 −2.09282 −1.51373 1.51373
−2.33152 −2.30901 3.43596 −3.37112 5.38349 0 −3.34797 2.33152 7.85981
1.2 −2.33152 2.30901 3.43596 3.37112 −5.38349 0 −3.34797 2.33152 −7.85981
1.3 −1.52077 −2.12621 0.312752 0.589391 3.23349 0 2.56592 1.52077 −0.896331
1.4 −1.52077 2.12621 0.312752 −0.589391 −3.23349 0 2.56592 1.52077 0.896331
1.5 −0.579810 −1.89204 −1.66382 1.47362 1.09702 0 2.12432 0.579810 −0.854419
1.6 −0.579810 1.89204 −1.66382 −1.47362 −1.09702 0 2.12432 0.579810 0.854419
1.7 2.43210 −0.753592 3.91511 −0.341537 −1.83281 0 4.65773 −2.43210 −0.830652
1.8 2.43210 0.753592 3.91511 0.341537 1.83281 0 4.65773 −2.43210 0.830652
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$
$$13$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.ci 8
7.b odd 2 1 inner 8281.2.a.ci 8
13.b even 2 1 8281.2.a.cl 8
13.c even 3 2 637.2.f.l 16
91.b odd 2 1 8281.2.a.cl 8
91.g even 3 2 637.2.g.m 16
91.h even 3 2 637.2.h.m 16
91.m odd 6 2 637.2.g.m 16
91.n odd 6 2 637.2.f.l 16
91.v odd 6 2 637.2.h.m 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.l 16 13.c even 3 2
637.2.f.l 16 91.n odd 6 2
637.2.g.m 16 91.g even 3 2
637.2.g.m 16 91.m odd 6 2
637.2.h.m 16 91.h even 3 2
637.2.h.m 16 91.v odd 6 2
8281.2.a.ci 8 1.a even 1 1 trivial
8281.2.a.ci 8 7.b odd 2 1 inner
8281.2.a.cl 8 13.b even 2 1
8281.2.a.cl 8 91.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 2 T_{2}^{3} - 5 T_{2}^{2} - 12 T_{2} - 5$$ $$T_{3}^{8} - 14 T_{3}^{6} + 67 T_{3}^{4} - 120 T_{3}^{2} + 49$$ $$T_{5}^{8} - 14 T_{5}^{6} + 31 T_{5}^{4} - 12 T_{5}^{2} + 1$$ $$T_{11}^{4} - 2 T_{11}^{3} - 5 T_{11}^{2} + 12 T_{11} - 5$$ $$T_{17}^{8} - 58 T_{17}^{6} + 966 T_{17}^{4} - 3866 T_{17}^{2} + 3721$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -5 - 12 T - 5 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$3$ $$49 - 120 T^{2} + 67 T^{4} - 14 T^{6} + T^{8}$$
$5$ $$1 - 12 T^{2} + 31 T^{4} - 14 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$( -5 + 12 T - 5 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$3721 - 3866 T^{2} + 966 T^{4} - 58 T^{6} + T^{8}$$
$19$ $$1225 - 6252 T^{2} + 2311 T^{4} - 94 T^{6} + T^{8}$$
$23$ $$( 100 - 56 T - 14 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$29$ $$( 1781 - 190 T - 85 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$26569 - 11726 T^{2} + 1518 T^{4} - 70 T^{6} + T^{8}$$
$37$ $$( -380 + 396 T - 86 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$4096 - 7168 T^{2} + 3008 T^{4} - 176 T^{6} + T^{8}$$
$43$ $$( -3205 - 1044 T - 15 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$47$ $$27889 - 131954 T^{2} + 13374 T^{4} - 226 T^{6} + T^{8}$$
$53$ $$( 271 - 70 T - 80 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$59$ $$169 - 3346 T^{2} + 7494 T^{4} - 194 T^{6} + T^{8}$$
$61$ $$19600 - 18992 T^{2} + 3196 T^{4} - 108 T^{6} + T^{8}$$
$67$ $$( -283 - 222 T - 6 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$71$ $$( 5956 - 268 T - 166 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$73$ $$2226064 - 706640 T^{2} + 45484 T^{4} - 428 T^{6} + T^{8}$$
$79$ $$( 8164 - 712 T - 278 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$83$ $$405769 - 257574 T^{2} + 27838 T^{4} - 350 T^{6} + T^{8}$$
$89$ $$45225625 - 8007708 T^{2} + 119687 T^{4} - 606 T^{6} + T^{8}$$
$97$ $$310249 - 355846 T^{2} + 30631 T^{4} - 364 T^{6} + T^{8}$$