Properties

Label 8281.2.a.bx
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.746052.1
Defining polynomial: \(x^{5} - x^{4} - 7 x^{3} + 8 x + 2\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + \beta_{4} q^{3} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{6} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + \beta_{4} q^{3} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{6} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{8} + ( 1 + \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{10} + ( -2 + \beta_{3} ) q^{11} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{12} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} ) q^{15} + ( 2 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} ) q^{17} + ( -2 - \beta_{2} + \beta_{3} ) q^{18} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{19} + ( 1 - \beta_{1} + 3 \beta_{3} ) q^{20} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{22} + ( 2 - \beta_{4} ) q^{23} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{24} + ( 1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{25} + ( 2 \beta_{1} + \beta_{2} ) q^{27} + ( -1 + 2 \beta_{1} + \beta_{4} ) q^{29} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} ) q^{30} + ( -2 - 2 \beta_{2} - \beta_{4} ) q^{31} + ( -5 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{32} + ( 2 - 2 \beta_{1} - 3 \beta_{4} ) q^{33} + ( -5 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{34} + ( 1 - 2 \beta_{3} - \beta_{4} ) q^{36} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{37} + ( 2 + \beta_{1} + \beta_{3} - 4 \beta_{4} ) q^{38} + ( -6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{40} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{41} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{2} + 3 \beta_{4} ) q^{44} + ( -6 + 2 \beta_{3} - 2 \beta_{4} ) q^{45} + ( -1 + 3 \beta_{1} + \beta_{3} + \beta_{4} ) q^{46} + ( -\beta_{3} + 4 \beta_{4} ) q^{47} + ( 5 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{48} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{50} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{51} + ( 3 + 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{53} + ( 5 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{54} + ( 2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{55} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{57} + ( 6 - 2 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{58} + ( 2 - \beta_{3} ) q^{59} + ( 7 - 5 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{60} + ( 3 + 4 \beta_{3} - 2 \beta_{4} ) q^{61} + ( 5 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{62} + ( 3 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{64} + ( -5 + 5 \beta_{1} + \beta_{3} + 5 \beta_{4} ) q^{66} + ( -4 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{67} + ( 8 - 7 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{68} + ( -4 - \beta_{2} + 2 \beta_{4} ) q^{69} + ( -2 - 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{71} + ( 4 + \beta_{3} + 3 \beta_{4} ) q^{72} + ( \beta_{2} - 2 \beta_{3} ) q^{73} + ( -7 + \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - \beta_{4} ) q^{74} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{75} + ( 1 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{76} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 11 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{80} + ( -5 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{81} + ( -8 + 8 \beta_{1} + \beta_{2} + 3 \beta_{4} ) q^{82} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{4} ) q^{83} + ( 4 - \beta_{2} + 4 \beta_{4} ) q^{85} + ( -6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{86} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{87} + ( -1 - 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{89} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{90} + ( 5 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{92} + ( -4 - 4 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} ) q^{93} + ( -4 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{94} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 6 \beta_{4} ) q^{95} + ( -9 + 9 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{96} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{97} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} - 9q^{8} + 3q^{9} + O(q^{10}) \) \( 5q - 4q^{2} + 8q^{4} + 2q^{5} - 5q^{6} - 9q^{8} + 3q^{9} + 5q^{10} - 11q^{11} - 5q^{12} + 10q^{16} + 5q^{17} - 9q^{18} + 9q^{19} + q^{20} + 8q^{22} + 10q^{23} + 9q^{25} - 3q^{29} - 13q^{30} - 6q^{31} - 22q^{32} + 8q^{33} - 22q^{34} + 7q^{36} - 4q^{37} + 10q^{38} - 28q^{40} + 14q^{41} + 2q^{43} - 32q^{45} - 3q^{46} + q^{47} + 23q^{48} - 9q^{50} - 8q^{51} + 17q^{53} + 23q^{54} + 16q^{57} + 27q^{58} + 11q^{59} + 29q^{60} + 11q^{61} + 23q^{62} + 9q^{64} - 21q^{66} - 13q^{67} + 32q^{68} - 18q^{69} - 15q^{71} + 19q^{72} - 33q^{74} + 20q^{75} + 8q^{76} + 2q^{79} + 55q^{80} - 19q^{81} - 34q^{82} + 6q^{83} + 22q^{85} - 28q^{86} + 8q^{87} - 3q^{88} - 4q^{89} + 34q^{90} + 21q^{92} - 18q^{93} - 20q^{94} - 12q^{95} - 37q^{96} - 12q^{97} - 11q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 4 \nu + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 4 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-8 \beta_{4} + 9 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} + 19\)

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
−1.72525
−1.21332
−0.265608
1.19566
3.00852
−2.72525 1.34642 5.42699 2.18716 −3.66932 0 −9.33940 −1.18716 −5.96057
1.2 −2.21332 −2.47443 2.89879 −2.12280 5.47671 0 −1.98932 3.12280 4.69843
1.3 −1.26561 2.62728 −0.398235 −2.90260 −3.32511 0 3.03523 3.90260 3.67356
1.4 0.195656 0.259788 −1.96172 3.93251 0.0508292 0 −0.775135 −2.93251 0.769420
1.5 2.00852 −1.75906 2.03417 0.905722 −3.53311 0 0.0686323 0.0942784 1.81916
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.bx 5
7.b odd 2 1 8281.2.a.bw 5
7.d odd 6 2 1183.2.e.f 10
13.b even 2 1 637.2.a.k 5
39.d odd 2 1 5733.2.a.bm 5
91.b odd 2 1 637.2.a.l 5
91.r even 6 2 637.2.e.m 10
91.s odd 6 2 91.2.e.c 10
273.g even 2 1 5733.2.a.bl 5
273.ba even 6 2 819.2.j.h 10
364.x even 6 2 1456.2.r.p 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 91.s odd 6 2
637.2.a.k 5 13.b even 2 1
637.2.a.l 5 91.b odd 2 1
637.2.e.m 10 91.r even 6 2
819.2.j.h 10 273.ba even 6 2
1183.2.e.f 10 7.d odd 6 2
1456.2.r.p 10 364.x even 6 2
5733.2.a.bl 5 273.g even 2 1
5733.2.a.bm 5 39.d odd 2 1
8281.2.a.bw 5 7.b odd 2 1
8281.2.a.bx 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{5} + 4 T_{2}^{4} - T_{2}^{3} - 17 T_{2}^{2} - 12 T_{2} + 3 \)
\( T_{3}^{5} - 9 T_{3}^{3} + 16 T_{3} - 4 \)
\( T_{5}^{5} - 2 T_{5}^{4} - 15 T_{5}^{3} + 20 T_{5}^{2} + 48 T_{5} - 48 \)
\( T_{11}^{5} + 11 T_{11}^{4} + 36 T_{11}^{3} + 22 T_{11}^{2} - 45 T_{11} - 33 \)
\( T_{17}^{5} - 5 T_{17}^{4} - 22 T_{17}^{3} + 106 T_{17}^{2} + 93 T_{17} - 429 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 - 12 T - 17 T^{2} - T^{3} + 4 T^{4} + T^{5} \)
$3$ \( -4 + 16 T - 9 T^{3} + T^{5} \)
$5$ \( -48 + 48 T + 20 T^{2} - 15 T^{3} - 2 T^{4} + T^{5} \)
$7$ \( T^{5} \)
$11$ \( -33 - 45 T + 22 T^{2} + 36 T^{3} + 11 T^{4} + T^{5} \)
$13$ \( T^{5} \)
$17$ \( -429 + 93 T + 106 T^{2} - 22 T^{3} - 5 T^{4} + T^{5} \)
$19$ \( -223 + 173 T + 176 T^{2} - 14 T^{3} - 9 T^{4} + T^{5} \)
$23$ \( 12 - 12 T - 26 T^{2} + 31 T^{3} - 10 T^{4} + T^{5} \)
$29$ \( -108 + 144 T - 19 T^{2} - 25 T^{3} + 3 T^{4} + T^{5} \)
$31$ \( -356 + 508 T - 102 T^{2} - 61 T^{3} + 6 T^{4} + T^{5} \)
$37$ \( 7036 + 660 T - 678 T^{2} - 111 T^{3} + 4 T^{4} + T^{5} \)
$41$ \( -1584 - 2544 T + 940 T^{2} - 28 T^{3} - 14 T^{4} + T^{5} \)
$43$ \( 64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} \)
$47$ \( -5169 + 2811 T + 26 T^{2} - 124 T^{3} - T^{4} + T^{5} \)
$53$ \( 19959 - 12759 T + 2426 T^{2} - 74 T^{3} - 17 T^{4} + T^{5} \)
$59$ \( 33 - 45 T - 22 T^{2} + 36 T^{3} - 11 T^{4} + T^{5} \)
$61$ \( 8461 + 5881 T + 766 T^{2} - 122 T^{3} - 11 T^{4} + T^{5} \)
$67$ \( 22699 - 591 T - 2160 T^{2} - 162 T^{3} + 13 T^{4} + T^{5} \)
$71$ \( 6336 - 456 T - 853 T^{2} - 25 T^{3} + 15 T^{4} + T^{5} \)
$73$ \( -712 + 700 T + 42 T^{2} - 75 T^{3} + T^{5} \)
$79$ \( -1000 + 1500 T - 190 T^{2} - 137 T^{3} - 2 T^{4} + T^{5} \)
$83$ \( -7488 + 2688 T + 308 T^{2} - 124 T^{3} - 6 T^{4} + T^{5} \)
$89$ \( 7692 + 2148 T - 694 T^{2} - 155 T^{3} + 4 T^{4} + T^{5} \)
$97$ \( -2384 - 2240 T - 612 T^{2} - 16 T^{3} + 12 T^{4} + T^{5} \)
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