Properties

Label 8281.2.a.n
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 \beta q^{4} + ( 1 + \beta ) q^{5} + ( -2 - 3 \beta ) q^{6} + ( -1 - 4 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta ) q^{2} + ( 1 + \beta ) q^{3} + 3 \beta q^{4} + ( 1 + \beta ) q^{5} + ( -2 - 3 \beta ) q^{6} + ( -1 - 4 \beta ) q^{8} + ( -1 + 3 \beta ) q^{9} + ( -2 - 3 \beta ) q^{10} + ( 3 - 3 \beta ) q^{11} + ( 3 + 6 \beta ) q^{12} + ( 2 + 3 \beta ) q^{15} + ( 5 + 3 \beta ) q^{16} + ( -5 + 4 \beta ) q^{17} + ( -2 - 5 \beta ) q^{18} + ( 3 - 3 \beta ) q^{19} + ( 3 + 6 \beta ) q^{20} + 3 \beta q^{22} + ( 2 - 4 \beta ) q^{23} + ( -5 - 9 \beta ) q^{24} + ( -3 + 3 \beta ) q^{25} + ( -1 + 2 \beta ) q^{27} + ( -1 + 5 \beta ) q^{29} + ( -5 - 8 \beta ) q^{30} + ( 5 - 6 \beta ) q^{31} + ( -6 - 3 \beta ) q^{32} -3 \beta q^{33} + ( 1 - 3 \beta ) q^{34} + ( 9 + 6 \beta ) q^{36} -4 q^{37} + 3 \beta q^{38} + ( -5 - 9 \beta ) q^{40} + ( 4 - 2 \beta ) q^{41} + ( -2 + 9 \beta ) q^{43} -9 q^{44} + ( 2 + 5 \beta ) q^{45} + ( 2 + 6 \beta ) q^{46} + ( 1 - 2 \beta ) q^{47} + ( 8 + 11 \beta ) q^{48} -3 \beta q^{50} + ( -1 + 3 \beta ) q^{51} + ( 7 - 2 \beta ) q^{53} + ( -1 - 3 \beta ) q^{54} -3 \beta q^{55} -3 \beta q^{57} + ( -4 - 9 \beta ) q^{58} + ( 1 - 2 \beta ) q^{59} + ( 9 + 15 \beta ) q^{60} + 6 q^{61} + ( 1 + 7 \beta ) q^{62} + ( -1 + 6 \beta ) q^{64} + ( 3 + 6 \beta ) q^{66} + ( 3 + 6 \beta ) q^{67} + ( 12 - 3 \beta ) q^{68} + ( -2 - 6 \beta ) q^{69} + ( -2 + 10 \beta ) q^{71} + ( -11 - 11 \beta ) q^{72} -2 q^{73} + ( 4 + 4 \beta ) q^{74} + 3 \beta q^{75} -9 q^{76} + 4 q^{79} + ( 8 + 11 \beta ) q^{80} + ( 4 - 6 \beta ) q^{81} -2 q^{82} + ( -3 + 6 \beta ) q^{83} + ( -1 + 3 \beta ) q^{85} + ( -7 - 16 \beta ) q^{86} + ( 4 + 9 \beta ) q^{87} + ( 9 + 3 \beta ) q^{88} + ( 13 - 5 \beta ) q^{89} + ( -7 - 12 \beta ) q^{90} + ( -12 - 6 \beta ) q^{92} + ( -1 - 7 \beta ) q^{93} + ( 1 + 3 \beta ) q^{94} -3 \beta q^{95} + ( -9 - 12 \beta ) q^{96} + ( 14 + 3 \beta ) q^{97} + ( -12 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 7q^{6} - 6q^{8} + q^{9} + O(q^{10}) \) \( 2q - 3q^{2} + 3q^{3} + 3q^{4} + 3q^{5} - 7q^{6} - 6q^{8} + q^{9} - 7q^{10} + 3q^{11} + 12q^{12} + 7q^{15} + 13q^{16} - 6q^{17} - 9q^{18} + 3q^{19} + 12q^{20} + 3q^{22} - 19q^{24} - 3q^{25} + 3q^{29} - 18q^{30} + 4q^{31} - 15q^{32} - 3q^{33} - q^{34} + 24q^{36} - 8q^{37} + 3q^{38} - 19q^{40} + 6q^{41} + 5q^{43} - 18q^{44} + 9q^{45} + 10q^{46} + 27q^{48} - 3q^{50} + q^{51} + 12q^{53} - 5q^{54} - 3q^{55} - 3q^{57} - 17q^{58} + 33q^{60} + 12q^{61} + 9q^{62} + 4q^{64} + 12q^{66} + 12q^{67} + 21q^{68} - 10q^{69} + 6q^{71} - 33q^{72} - 4q^{73} + 12q^{74} + 3q^{75} - 18q^{76} + 8q^{79} + 27q^{80} + 2q^{81} - 4q^{82} + q^{85} - 30q^{86} + 17q^{87} + 21q^{88} + 21q^{89} - 26q^{90} - 30q^{92} - 9q^{93} + 5q^{94} - 3q^{95} - 30q^{96} + 31q^{97} - 21q^{99} + O(q^{100}) \)

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
1.61803
−0.618034
−2.61803 2.61803 4.85410 2.61803 −6.85410 0 −7.47214 3.85410 −6.85410
1.2 −0.381966 0.381966 −1.85410 0.381966 −0.145898 0 1.47214 −2.85410 −0.145898
\(n\): e.g. 2-40 or 990-1000
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.n 2
7.b odd 2 1 1183.2.a.c 2
13.b even 2 1 8281.2.a.bb 2
13.e even 6 2 637.2.f.c 4
91.b odd 2 1 1183.2.a.g 2
91.i even 4 2 1183.2.c.c 4
91.k even 6 2 637.2.h.f 4
91.l odd 6 2 637.2.h.g 4
91.p odd 6 2 637.2.g.b 4
91.t odd 6 2 91.2.f.a 4
91.u even 6 2 637.2.g.c 4
273.u even 6 2 819.2.o.c 4
364.bc even 6 2 1456.2.s.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 91.t odd 6 2
637.2.f.c 4 13.e even 6 2
637.2.g.b 4 91.p odd 6 2
637.2.g.c 4 91.u even 6 2
637.2.h.f 4 91.k even 6 2
637.2.h.g 4 91.l odd 6 2
819.2.o.c 4 273.u even 6 2
1183.2.a.c 2 7.b odd 2 1
1183.2.a.g 2 91.b odd 2 1
1183.2.c.c 4 91.i even 4 2
1456.2.s.h 4 364.bc even 6 2
8281.2.a.n 2 1.a even 1 1 trivial
8281.2.a.bb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} + 3 T_{2} + 1 \)
\( T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{5}^{2} - 3 T_{5} + 1 \)
\( T_{11}^{2} - 3 T_{11} - 9 \)
\( T_{17}^{2} + 6 T_{17} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + T^{2} \)
$3$ \( 1 - 3 T + T^{2} \)
$5$ \( 1 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -9 - 3 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -11 + 6 T + T^{2} \)
$19$ \( -9 - 3 T + T^{2} \)
$23$ \( -20 + T^{2} \)
$29$ \( -29 - 3 T + T^{2} \)
$31$ \( -41 - 4 T + T^{2} \)
$37$ \( ( 4 + T )^{2} \)
$41$ \( 4 - 6 T + T^{2} \)
$43$ \( -95 - 5 T + T^{2} \)
$47$ \( -5 + T^{2} \)
$53$ \( 31 - 12 T + T^{2} \)
$59$ \( -5 + T^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( -9 - 12 T + T^{2} \)
$71$ \( -116 - 6 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( -4 + T )^{2} \)
$83$ \( -45 + T^{2} \)
$89$ \( 79 - 21 T + T^{2} \)
$97$ \( 229 - 31 T + T^{2} \)
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