Properties

Label 8085.2.a.bk
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + \beta_{1} q^{6} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} + q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + \beta_{1} q^{6} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{8} + q^{9} + \beta_{1} q^{10} + q^{11} + ( -1 - \beta_{1} - \beta_{2} ) q^{12} + ( \beta_{1} + \beta_{2} ) q^{13} + q^{15} + ( 3 + 4 \beta_{1} ) q^{16} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} + ( -2 - 2 \beta_{2} ) q^{19} + ( -1 - \beta_{1} - \beta_{2} ) q^{20} -\beta_{1} q^{22} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{24} + q^{25} + ( -2 - 3 \beta_{1} - \beta_{2} ) q^{26} - q^{27} + ( -4 + 2 \beta_{1} ) q^{29} -\beta_{1} q^{30} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} + ( -8 - 3 \beta_{1} - 2 \beta_{2} ) q^{32} - q^{33} + ( 8 + 3 \beta_{1} + 3 \beta_{2} ) q^{34} + ( 1 + \beta_{1} + \beta_{2} ) q^{36} -2 q^{37} + ( -2 + 6 \beta_{1} ) q^{38} + ( -\beta_{1} - \beta_{2} ) q^{39} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{40} + ( 4 + 2 \beta_{1} ) q^{41} + ( \beta_{1} + 3 \beta_{2} ) q^{43} + ( 1 + \beta_{1} + \beta_{2} ) q^{44} - q^{45} + ( 8 - 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -3 - 4 \beta_{1} ) q^{48} -\beta_{1} q^{50} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{51} + ( 8 + 5 \beta_{1} + \beta_{2} ) q^{52} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{53} + \beta_{1} q^{54} - q^{55} + ( 2 + 2 \beta_{2} ) q^{57} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 1 + \beta_{1} + \beta_{2} ) q^{60} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{62} + ( 1 + 7 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -\beta_{1} - \beta_{2} ) q^{65} + \beta_{1} q^{66} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -10 - 11 \beta_{1} - \beta_{2} ) q^{68} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{72} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{73} + 2 \beta_{1} q^{74} - q^{75} + ( -14 - 4 \beta_{1} - 2 \beta_{2} ) q^{76} + ( 2 + 3 \beta_{1} + \beta_{2} ) q^{78} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -3 - 4 \beta_{1} ) q^{80} + q^{81} + ( -6 - 6 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -2 + 3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -2 + 3 \beta_{1} + \beta_{2} ) q^{85} + ( -7 \beta_{1} - \beta_{2} ) q^{86} + ( 4 - 2 \beta_{1} ) q^{87} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 + 4 \beta_{1} ) q^{89} + \beta_{1} q^{90} + ( 8 - 6 \beta_{1} - 2 \beta_{2} ) q^{92} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{94} + ( 2 + 2 \beta_{2} ) q^{95} + ( 8 + 3 \beta_{1} + 2 \beta_{2} ) q^{96} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - 2 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 2 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{1} + 2\)

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
2.17009
−1.48119
0.311108
−2.70928 −1.00000 5.34017 −1.00000 2.70928 0 −9.04945 1.00000 2.70928
1.2 −0.193937 −1.00000 −1.96239 −1.00000 0.193937 0 0.768452 1.00000 0.193937
1.3 1.90321 −1.00000 1.62222 −1.00000 −1.90321 0 −0.719004 1.00000 −1.90321
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 8085.2.a.bk 3
7.b odd 2 1 165.2.a.c 3
21.c even 2 1 495.2.a.e 3
28.d even 2 1 2640.2.a.be 3
35.c odd 2 1 825.2.a.k 3
35.f even 4 2 825.2.c.g 6
77.b even 2 1 1815.2.a.m 3
84.h odd 2 1 7920.2.a.cj 3
105.g even 2 1 2475.2.a.bb 3
105.k odd 4 2 2475.2.c.r 6
231.h odd 2 1 5445.2.a.z 3
385.h even 2 1 9075.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.c 3 7.b odd 2 1
495.2.a.e 3 21.c even 2 1
825.2.a.k 3 35.c odd 2 1
825.2.c.g 6 35.f even 4 2
1815.2.a.m 3 77.b even 2 1
2475.2.a.bb 3 105.g even 2 1
2475.2.c.r 6 105.k odd 4 2
2640.2.a.be 3 28.d even 2 1
5445.2.a.z 3 231.h odd 2 1
7920.2.a.cj 3 84.h odd 2 1
8085.2.a.bk 3 1.a even 1 1 trivial
9075.2.a.cf 3 385.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{3} + T_{2}^{2} - 5 T_{2} - 1 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 12 T_{13} + 8 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 52 T_{17} + 184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 5 T + T^{2} + T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$17$ \( 184 - 52 T - 2 T^{2} + T^{3} \)
$19$ \( -160 - 16 T + 8 T^{2} + T^{3} \)
$23$ \( -128 - 64 T + T^{3} \)
$29$ \( -40 + 12 T + 10 T^{2} + T^{3} \)
$31$ \( -128 - 32 T + 8 T^{2} + T^{3} \)
$37$ \( ( 2 + T )^{3} \)
$41$ \( -8 + 44 T - 14 T^{2} + T^{3} \)
$43$ \( 400 - 80 T - 4 T^{2} + T^{3} \)
$47$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$53$ \( 8 - 52 T + 6 T^{2} + T^{3} \)
$59$ \( -320 - 16 T + 12 T^{2} + T^{3} \)
$61$ \( 248 - 52 T - 6 T^{2} + T^{3} \)
$67$ \( -64 - 48 T + 4 T^{2} + T^{3} \)
$71$ \( 128 - 32 T - 8 T^{2} + T^{3} \)
$73$ \( 344 + 4 T - 14 T^{2} + T^{3} \)
$79$ \( 800 - 64 T - 12 T^{2} + T^{3} \)
$83$ \( -16 - 120 T + T^{3} \)
$89$ \( 200 - 52 T - 10 T^{2} + T^{3} \)
$97$ \( 8 + 108 T + 22 T^{2} + T^{3} \)
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