# Properties

 Label 8281.2.a.by Level $8281$ Weight $2$ Character orbit 8281.a Self dual yes Analytic conductor $66.124$ Analytic rank $1$ 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7674048.1 Defining polynomial: $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + \beta_{4} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{2} - \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + \beta_{4} q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + ( 1 + \beta_{2} - \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{6} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{5} ) q^{10} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{11} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{12} + ( -3 - \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{15} + ( 2 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{16} + ( 1 - \beta_{4} + \beta_{5} ) q^{17} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{18} + ( -2 \beta_{2} - \beta_{5} ) q^{19} + ( 6 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{20} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{22} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{23} + ( -2 \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{24} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{25} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{27} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{29} + ( 3 - 3 \beta_{1} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{30} + ( -3 - \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{31} + ( -2 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{32} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{33} + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{34} + ( -1 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{36} + ( -3 - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{37} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{38} + ( -8 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{40} + ( 5 - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{43} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{45} + ( -2 - 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} ) q^{46} + ( 3 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{48} + ( -7 + 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{50} + ( -4 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{51} + ( -3 - 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 2 + 2 \beta_{4} - \beta_{5} ) q^{54} + ( 1 - 5 \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{55} + ( 1 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{57} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{58} + ( 1 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{59} + ( -2 + 3 \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{60} + ( 2 + 3 \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{61} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{62} + ( 3 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{64} + ( 3 + 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{66} + ( -5 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{67} + ( -1 + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{68} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 1 - 7 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{72} + ( 5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{73} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{74} + ( -7 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{75} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -4 + 4 \beta_{2} + 2 \beta_{5} ) q^{79} + ( 6 - 4 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{80} + ( -\beta_{1} - 3 \beta_{3} + 3 \beta_{4} ) q^{81} + ( -4 + 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{82} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{83} + ( 3 + \beta_{1} + 4 \beta_{3} - 4 \beta_{4} ) q^{85} + ( 5 - 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{86} + ( -1 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{87} + ( -4 + 5 \beta_{1} - \beta_{2} - 4 \beta_{4} ) q^{88} + ( 6 + 2 \beta_{4} - 3 \beta_{5} ) q^{89} + ( 1 + 2 \beta_{2} - 6 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{90} + ( 2 - 4 \beta_{3} + 2 \beta_{4} ) q^{92} + ( 2 - 5 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{94} + ( -5 + 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{95} + ( -7 - \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{96} + ( 5 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{97} + ( -1 - 3 \beta_{1} + 3 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 12 q^{8} + 4 q^{9} + O(q^{10})$$ $$6 q - 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 12 q^{8} + 4 q^{9} - 12 q^{10} - 4 q^{11} - 2 q^{12} - 20 q^{15} + 8 q^{16} + 4 q^{17} + 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 10 q^{25} - 6 q^{27} - 8 q^{29} + 8 q^{30} - 14 q^{31} - 8 q^{32} + 16 q^{33} - 2 q^{34} - 10 q^{36} - 12 q^{37} + 2 q^{38} - 46 q^{40} + 28 q^{41} + 2 q^{43} + 20 q^{44} + 16 q^{45} - 20 q^{46} + 14 q^{47} - 2 q^{48} - 32 q^{50} - 26 q^{51} - 22 q^{53} + 14 q^{54} - 6 q^{55} - 4 q^{58} - 2 q^{59} + 14 q^{61} + 4 q^{62} + 26 q^{64} + 26 q^{66} - 24 q^{67} - 8 q^{68} - 4 q^{69} - 4 q^{71} - 8 q^{72} + 36 q^{73} - 6 q^{74} - 46 q^{75} - 26 q^{76} - 28 q^{79} + 36 q^{80} - 2 q^{81} - 14 q^{82} + 26 q^{83} + 20 q^{85} + 24 q^{86} - 2 q^{87} - 14 q^{88} + 42 q^{89} + 12 q^{90} + 12 q^{92} + 4 q^{94} - 22 q^{95} - 42 q^{96} + 24 q^{97} - 16 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8$$

## 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
 −1.70320 −1.10939 −0.276564 0.879640 1.82356 2.38595
−2.70320 0.345949 5.30727 3.25812 −0.935168 0 −8.94020 −2.88032 −8.80735
1.2 −2.10939 −2.26165 2.44952 3.60178 4.77070 0 −0.948212 2.11505 −7.59755
1.3 −1.27656 1.16793 −0.370384 −1.81487 −1.49093 0 3.02595 −1.63595 2.31680
1.4 −0.120360 0.582292 −1.98551 −1.68817 −0.0700846 0 0.479696 −2.66094 0.203187
1.5 0.823556 −2.66029 −1.32176 3.16209 −2.19090 0 −2.73565 4.07715 2.60416
1.6 1.38595 2.82577 −0.0791355 −0.518957 3.91639 0 −2.88158 4.98500 −0.719250
 $$n$$: 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.by 6
7.b odd 2 1 1183.2.a.m 6
13.b even 2 1 8281.2.a.ch 6
13.f odd 12 2 637.2.q.h 12
91.b odd 2 1 1183.2.a.p 6
91.i even 4 2 1183.2.c.i 12
91.w even 12 2 637.2.u.h 12
91.x odd 12 2 637.2.k.g 12
91.ba even 12 2 637.2.k.h 12
91.bc even 12 2 91.2.q.a 12
91.bd odd 12 2 637.2.u.i 12
273.ca odd 12 2 819.2.ct.a 12
364.bv odd 12 2 1456.2.cc.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.q.a 12 91.bc even 12 2
637.2.k.g 12 91.x odd 12 2
637.2.k.h 12 91.ba even 12 2
637.2.q.h 12 13.f odd 12 2
637.2.u.h 12 91.w even 12 2
637.2.u.i 12 91.bd odd 12 2
819.2.ct.a 12 273.ca odd 12 2
1183.2.a.m 6 7.b odd 2 1
1183.2.a.p 6 91.b odd 2 1
1183.2.c.i 12 91.i even 4 2
1456.2.cc.c 12 364.bv odd 12 2
8281.2.a.by 6 1.a even 1 1 trivial
8281.2.a.ch 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8281))$$:

 $$T_{2}^{6} + 4 T_{2}^{5} - 12 T_{2}^{3} - 4 T_{2}^{2} + 8 T_{2} + 1$$ $$T_{3}^{6} - 11 T_{3}^{4} + 2 T_{3}^{3} + 25 T_{3}^{2} - 20 T_{3} + 4$$ $$T_{5}^{6} - 6 T_{5}^{5} - 2 T_{5}^{4} + 50 T_{5}^{3} - 2 T_{5}^{2} - 128 T_{5} - 59$$ $$T_{11}^{6} + 4 T_{11}^{5} - 17 T_{11}^{4} - 16 T_{11}^{3} + 85 T_{11}^{2} - 72 T_{11} + 16$$ $$T_{17}^{6} - 4 T_{17}^{5} - 21 T_{17}^{4} + 60 T_{17}^{3} + 167 T_{17}^{2} - 224 T_{17} - 491$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T - 4 T^{2} - 12 T^{3} + 4 T^{5} + T^{6}$$
$3$ $$4 - 20 T + 25 T^{2} + 2 T^{3} - 11 T^{4} + T^{6}$$
$5$ $$-59 - 128 T - 2 T^{2} + 50 T^{3} - 2 T^{4} - 6 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$16 - 72 T + 85 T^{2} - 16 T^{3} - 17 T^{4} + 4 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$-491 - 224 T + 167 T^{2} + 60 T^{3} - 21 T^{4} - 4 T^{5} + T^{6}$$
$19$ $$-236 + 32 T + 149 T^{2} - 27 T^{4} - 2 T^{5} + T^{6}$$
$23$ $$6208 + 1472 T - 1616 T^{2} - 608 T^{3} - 20 T^{4} + 12 T^{5} + T^{6}$$
$29$ $$3169 + 336 T - 1412 T^{2} - 566 T^{3} - 44 T^{4} + 8 T^{5} + T^{6}$$
$31$ $$-956 + 1360 T - 295 T^{2} - 198 T^{3} + 30 T^{4} + 14 T^{5} + T^{6}$$
$37$ $$-41904 + 22464 T + 2916 T^{2} - 1044 T^{3} - 87 T^{4} + 12 T^{5} + T^{6}$$
$41$ $$-29744 + 20224 T - 3228 T^{2} - 636 T^{3} + 257 T^{4} - 28 T^{5} + T^{6}$$
$43$ $$1552 + 3776 T + 2421 T^{2} + 90 T^{3} - 109 T^{4} - 2 T^{5} + T^{6}$$
$47$ $$3076 - 6696 T - 443 T^{2} + 758 T^{3} - 38 T^{4} - 14 T^{5} + T^{6}$$
$53$ $$-2339 + 7302 T - 3353 T^{2} - 700 T^{3} + 91 T^{4} + 22 T^{5} + T^{6}$$
$59$ $$-67616 + 7192 T + 6845 T^{2} - 210 T^{3} - 162 T^{4} + 2 T^{5} + T^{6}$$
$61$ $$2368 - 1600 T - 1888 T^{2} + 1416 T^{3} - 87 T^{4} - 14 T^{5} + T^{6}$$
$67$ $$24772 - 19256 T - 11855 T^{2} - 1408 T^{3} + 94 T^{4} + 24 T^{5} + T^{6}$$
$71$ $$-6848 + 2496 T + 1072 T^{2} - 256 T^{3} - 68 T^{4} + 4 T^{5} + T^{6}$$
$73$ $$-37232 + 12112 T + 4924 T^{2} - 2788 T^{3} + 481 T^{4} - 36 T^{5} + T^{6}$$
$79$ $$-512 + 1664 T - 1584 T^{2} + 192 T^{3} + 212 T^{4} + 28 T^{5} + T^{6}$$
$83$ $$-11888 + 25544 T - 8335 T^{2} + 330 T^{3} + 186 T^{4} - 26 T^{5} + T^{6}$$
$89$ $$42832 + 28352 T - 10775 T^{2} - 1640 T^{3} + 553 T^{4} - 42 T^{5} + T^{6}$$
$97$ $$7312 - 14224 T - 3815 T^{2} + 934 T^{3} + 97 T^{4} - 24 T^{5} + T^{6}$$