Properties

Label 8085.2.a.ba
Level $8085$
Weight $2$
Character orbit 8085.a
Self dual yes
Analytic conductor $64.559$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( -1 + \beta ) q^{6} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + q^{3} + ( 1 - 2 \beta ) q^{4} + q^{5} + ( -1 + \beta ) q^{6} + ( -3 + \beta ) q^{8} + q^{9} + ( -1 + \beta ) q^{10} - q^{11} + ( 1 - 2 \beta ) q^{12} -4 \beta q^{13} + q^{15} + 3 q^{16} + ( 4 + 2 \beta ) q^{17} + ( -1 + \beta ) q^{18} + ( 4 - 2 \beta ) q^{19} + ( 1 - 2 \beta ) q^{20} + ( 1 - \beta ) q^{22} -4 q^{23} + ( -3 + \beta ) q^{24} + q^{25} + ( -8 + 4 \beta ) q^{26} + q^{27} + ( -2 + 2 \beta ) q^{29} + ( -1 + \beta ) q^{30} + ( 3 + \beta ) q^{32} - q^{33} + 2 \beta q^{34} + ( 1 - 2 \beta ) q^{36} + ( 6 - 4 \beta ) q^{37} + ( -8 + 6 \beta ) q^{38} -4 \beta q^{39} + ( -3 + \beta ) q^{40} + ( -2 + 2 \beta ) q^{41} + ( -6 + 2 \beta ) q^{43} + ( -1 + 2 \beta ) q^{44} + q^{45} + ( 4 - 4 \beta ) q^{46} + 4 q^{47} + 3 q^{48} + ( -1 + \beta ) q^{50} + ( 4 + 2 \beta ) q^{51} + ( 16 - 4 \beta ) q^{52} + ( -2 - 8 \beta ) q^{53} + ( -1 + \beta ) q^{54} - q^{55} + ( 4 - 2 \beta ) q^{57} + ( 6 - 4 \beta ) q^{58} + 4 q^{59} + ( 1 - 2 \beta ) q^{60} + ( 6 - 4 \beta ) q^{61} + ( -7 + 2 \beta ) q^{64} -4 \beta q^{65} + ( 1 - \beta ) q^{66} + 4 \beta q^{67} + ( -4 - 6 \beta ) q^{68} -4 q^{69} + ( 8 + 4 \beta ) q^{71} + ( -3 + \beta ) q^{72} + 8 \beta q^{73} + ( -14 + 10 \beta ) q^{74} + q^{75} + ( 12 - 10 \beta ) q^{76} + ( -8 + 4 \beta ) q^{78} -6 \beta q^{79} + 3 q^{80} + q^{81} + ( 6 - 4 \beta ) q^{82} + 10 q^{83} + ( 4 + 2 \beta ) q^{85} + ( 10 - 8 \beta ) q^{86} + ( -2 + 2 \beta ) q^{87} + ( 3 - \beta ) q^{88} + ( 2 + 4 \beta ) q^{89} + ( -1 + \beta ) q^{90} + ( -4 + 8 \beta ) q^{92} + ( -4 + 4 \beta ) q^{94} + ( 4 - 2 \beta ) q^{95} + ( 3 + \beta ) q^{96} + ( -6 + 4 \beta ) q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} + 2q^{15} + 6q^{16} + 8q^{17} - 2q^{18} + 8q^{19} + 2q^{20} + 2q^{22} - 8q^{23} - 6q^{24} + 2q^{25} - 16q^{26} + 2q^{27} - 4q^{29} - 2q^{30} + 6q^{32} - 2q^{33} + 2q^{36} + 12q^{37} - 16q^{38} - 6q^{40} - 4q^{41} - 12q^{43} - 2q^{44} + 2q^{45} + 8q^{46} + 8q^{47} + 6q^{48} - 2q^{50} + 8q^{51} + 32q^{52} - 4q^{53} - 2q^{54} - 2q^{55} + 8q^{57} + 12q^{58} + 8q^{59} + 2q^{60} + 12q^{61} - 14q^{64} + 2q^{66} - 8q^{68} - 8q^{69} + 16q^{71} - 6q^{72} - 28q^{74} + 2q^{75} + 24q^{76} - 16q^{78} + 6q^{80} + 2q^{81} + 12q^{82} + 20q^{83} + 8q^{85} + 20q^{86} - 4q^{87} + 6q^{88} + 4q^{89} - 2q^{90} - 8q^{92} - 8q^{94} + 8q^{95} + 6q^{96} - 12q^{97} - 2q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 1.00000 3.82843 1.00000 −2.41421 0 −4.41421 1.00000 −2.41421
1.2 0.414214 1.00000 −1.82843 1.00000 0.414214 0 −1.58579 1.00000 0.414214
\(n\): 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.ba 2
7.b odd 2 1 165.2.a.a 2
21.c even 2 1 495.2.a.d 2
28.d even 2 1 2640.2.a.bb 2
35.c odd 2 1 825.2.a.g 2
35.f even 4 2 825.2.c.e 4
77.b even 2 1 1815.2.a.k 2
84.h odd 2 1 7920.2.a.cg 2
105.g even 2 1 2475.2.a.m 2
105.k odd 4 2 2475.2.c.m 4
231.h odd 2 1 5445.2.a.m 2
385.h even 2 1 9075.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 7.b odd 2 1
495.2.a.d 2 21.c even 2 1
825.2.a.g 2 35.c odd 2 1
825.2.c.e 4 35.f even 4 2
1815.2.a.k 2 77.b even 2 1
2475.2.a.m 2 105.g even 2 1
2475.2.c.m 4 105.k odd 4 2
2640.2.a.bb 2 28.d even 2 1
5445.2.a.m 2 231.h odd 2 1
7920.2.a.cg 2 84.h odd 2 1
8085.2.a.ba 2 1.a even 1 1 trivial
9075.2.a.v 2 385.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8085))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( 8 - 8 T + T^{2} \)
$23$ \( ( 4 + T )^{2} \)
$29$ \( -4 + 4 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4 - 12 T + T^{2} \)
$41$ \( -4 + 4 T + T^{2} \)
$43$ \( 28 + 12 T + T^{2} \)
$47$ \( ( -4 + T )^{2} \)
$53$ \( -124 + 4 T + T^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 4 - 12 T + T^{2} \)
$67$ \( -32 + T^{2} \)
$71$ \( 32 - 16 T + T^{2} \)
$73$ \( -128 + T^{2} \)
$79$ \( -72 + T^{2} \)
$83$ \( ( -10 + T )^{2} \)
$89$ \( -28 - 4 T + T^{2} \)
$97$ \( 4 + 12 T + T^{2} \)
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