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: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8} + q^{9} + \beta_1 q^{10} + q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{12} + (\beta_{2} + \beta_1) q^{13} + q^{15} + (4 \beta_1 + 3) q^{16} + ( - \beta_{2} - 3 \beta_1 + 2) q^{17} - \beta_1 q^{18} + ( - 2 \beta_{2} - 2) q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{20} - \beta_1 q^{22} + (2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{2} + 2 \beta_1 + 2) q^{24} + q^{25} + ( - \beta_{2} - 3 \beta_1 - 2) q^{26} - q^{27} + (2 \beta_1 - 4) q^{29} - \beta_1 q^{30} + (2 \beta_{2} + 2 \beta_1 - 4) q^{31} + ( - 2 \beta_{2} - 3 \beta_1 - 8) q^{32} - q^{33} + (3 \beta_{2} + 3 \beta_1 + 8) q^{34} + (\beta_{2} + \beta_1 + 1) q^{36} - 2 q^{37} + (6 \beta_1 - 2) q^{38} + ( - \beta_{2} - \beta_1) q^{39} + (\beta_{2} + 2 \beta_1 + 2) q^{40} + (2 \beta_1 + 4) q^{41} + (3 \beta_{2} + \beta_1) q^{43} + (\beta_{2} + \beta_1 + 1) q^{44} - q^{45} + (2 \beta_{2} - 2 \beta_1 + 8) q^{46} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{47} + ( - 4 \beta_1 - 3) q^{48} - \beta_1 q^{50} + (\beta_{2} + 3 \beta_1 - 2) q^{51} + (\beta_{2} + 5 \beta_1 + 8) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{53} + \beta_1 q^{54} - q^{55} + (2 \beta_{2} + 2) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{58} + (2 \beta_{2} - 2 \beta_1 - 4) q^{59} + (\beta_{2} + \beta_1 + 1) q^{60} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{61} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{62} + (3 \beta_{2} + 7 \beta_1 + 1) q^{64} + ( - \beta_{2} - \beta_1) q^{65} + \beta_1 q^{66} + ( - 2 \beta_{2} - 2 \beta_1) q^{67} + ( - \beta_{2} - 11 \beta_1 - 10) q^{68} + ( - 2 \beta_{2} + 2 \beta_1) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - \beta_{2} - 2 \beta_1 - 2) q^{72} + ( - \beta_{2} + 3 \beta_1 + 4) q^{73} + 2 \beta_1 q^{74} - q^{75} + ( - 2 \beta_{2} - 4 \beta_1 - 14) q^{76} + (\beta_{2} + 3 \beta_1 + 2) q^{78} + ( - 2 \beta_{2} - 4 \beta_1 + 6) q^{79} + ( - 4 \beta_1 - 3) q^{80} + q^{81} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{82} + (3 \beta_{2} + 3 \beta_1 - 2) q^{83} + (\beta_{2} + 3 \beta_1 - 2) q^{85} + ( - \beta_{2} - 7 \beta_1) q^{86} + ( - 2 \beta_1 + 4) q^{87} + ( - \beta_{2} - 2 \beta_1 - 2) q^{88} + (4 \beta_1 + 2) q^{89} + \beta_1 q^{90} + ( - 2 \beta_{2} - 6 \beta_1 + 8) q^{92} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{93} + (2 \beta_{2} + 2 \beta_1 + 4) q^{94} + (2 \beta_{2} + 2) q^{95} + (2 \beta_{2} + 3 \beta_1 + 8) q^{96} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 3 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 9 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} - 5 q^{12} + 2 q^{13} + 3 q^{15} + 13 q^{16} + 2 q^{17} - q^{18} - 8 q^{19} - 5 q^{20} - q^{22} + 9 q^{24} + 3 q^{25} - 10 q^{26} - 3 q^{27} - 10 q^{29} - q^{30} - 8 q^{31} - 29 q^{32} - 3 q^{33} + 30 q^{34} + 5 q^{36} - 6 q^{37} - 2 q^{39} + 9 q^{40} + 14 q^{41} + 4 q^{43} + 5 q^{44} - 3 q^{45} + 24 q^{46} + 8 q^{47} - 13 q^{48} - q^{50} - 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 3 q^{55} + 8 q^{57} - 18 q^{58} - 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} - 2 q^{65} + q^{66} - 4 q^{67} - 42 q^{68} + 8 q^{71} - 9 q^{72} + 14 q^{73} + 2 q^{74} - 3 q^{75} - 48 q^{76} + 10 q^{78} + 12 q^{79} - 13 q^{80} + 3 q^{81} - 26 q^{82} - 2 q^{85} - 8 q^{86} + 10 q^{87} - 9 q^{88} + 10 q^{89} + q^{90} + 16 q^{92} + 8 q^{93} + 16 q^{94} + 8 q^{95} + 29 q^{96} - 22 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 1 \) Copy content Toggle raw display
\( T_{13}^{3} - 2T_{13}^{2} - 12T_{13} + 8 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 52T_{17} + 184 \) Copy content Toggle raw display

Hecke characteristic polynomials

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