Properties

Label 8670.2.a.v
Level $8670$
Weight $2$
Character orbit 8670.a
Self dual yes
Analytic conductor $69.230$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8670.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.2302985525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 510)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} + q^{16} + q^{18} + 4q^{19} - q^{20} - 4q^{22} + q^{24} + q^{25} - 2q^{26} + q^{27} + 2q^{29} - q^{30} - 8q^{31} + q^{32} - 4q^{33} + q^{36} - 6q^{37} + 4q^{38} - 2q^{39} - q^{40} + 6q^{41} - 4q^{43} - 4q^{44} - q^{45} + q^{48} - 7q^{49} + q^{50} - 2q^{52} - 10q^{53} + q^{54} + 4q^{55} + 4q^{57} + 2q^{58} - 4q^{59} - q^{60} + 2q^{61} - 8q^{62} + q^{64} + 2q^{65} - 4q^{66} + 4q^{67} + q^{72} + 6q^{73} - 6q^{74} + q^{75} + 4q^{76} - 2q^{78} - 8q^{79} - q^{80} + q^{81} + 6q^{82} - 12q^{83} - 4q^{86} + 2q^{87} - 4q^{88} - 6q^{89} - q^{90} - 8q^{93} - 4q^{95} + q^{96} + 14q^{97} - 7q^{98} - 4q^{99} + O(q^{100}) \)

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
0
1.00000 1.00000 1.00000 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(17\) \(1\)

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 8670.2.a.v 1
17.b even 2 1 510.2.a.e 1
51.c odd 2 1 1530.2.a.b 1
68.d odd 2 1 4080.2.a.ba 1
85.c even 2 1 2550.2.a.l 1
85.g odd 4 2 2550.2.d.k 2
255.h odd 2 1 7650.2.a.bx 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.e 1 17.b even 2 1
1530.2.a.b 1 51.c odd 2 1
2550.2.a.l 1 85.c even 2 1
2550.2.d.k 2 85.g odd 4 2
4080.2.a.ba 1 68.d odd 2 1
7650.2.a.bx 1 255.h odd 2 1
8670.2.a.v 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13} + 2 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 2 + T \)
$17$ \( T \)
$19$ \( -4 + T \)
$23$ \( T \)
$29$ \( -2 + T \)
$31$ \( 8 + T \)
$37$ \( 6 + T \)
$41$ \( -6 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 4 + T \)
$61$ \( -2 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( 8 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( -14 + T \)
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