Properties

Label 8281.2.a.ce
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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: 6.6.6995813.1
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
0.435907
−2.04394
−0.874884
1.51235
2.33401
−0.363441
−1.85816 2.29407 1.45276 0.197362 −4.26275 0 1.01686 2.26275 −0.366731
1.2 −1.55469 −0.489252 0.417051 1.19151 0.760633 0 2.46099 −2.76063 −1.85243
1.3 0.268125 −1.14301 −1.92811 2.56175 −0.306470 0 −1.05323 −1.69353 0.686871
1.4 0.851125 0.661223 −1.27559 −3.44148 0.562784 0 −2.78793 −2.56278 −2.92913
1.5 1.90556 0.428448 1.63116 1.47313 0.816433 0 −0.702849 −2.81643 2.80714
1.6 2.38804 −2.75148 3.70272 −0.982280 −6.57063 0 4.06616 4.57063 −2.34572
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

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.ce 6
7.b odd 2 1 8281.2.a.cf 6
7.c even 3 2 1183.2.e.g 12
13.b even 2 1 8281.2.a.bz 6
13.e even 6 2 637.2.f.k 12
91.b odd 2 1 8281.2.a.ca 6
91.k even 6 2 91.2.h.b yes 12
91.l odd 6 2 637.2.h.l 12
91.p odd 6 2 637.2.g.l 12
91.r even 6 2 1183.2.e.h 12
91.t odd 6 2 637.2.f.j 12
91.u even 6 2 91.2.g.b 12
273.x odd 6 2 819.2.n.d 12
273.bp odd 6 2 819.2.s.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.b 12 91.u even 6 2
91.2.h.b yes 12 91.k even 6 2
637.2.f.j 12 91.t odd 6 2
637.2.f.k 12 13.e even 6 2
637.2.g.l 12 91.p odd 6 2
637.2.h.l 12 91.l odd 6 2
819.2.n.d 12 273.x odd 6 2
819.2.s.d 12 273.bp odd 6 2
1183.2.e.g 12 7.c even 3 2
1183.2.e.h 12 91.r even 6 2
8281.2.a.bz 6 13.b even 2 1
8281.2.a.ca 6 91.b odd 2 1
8281.2.a.ce 6 1.a even 1 1 trivial
8281.2.a.cf 6 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

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

Hecke characteristic polynomials

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