Properties

Label 8470.2.a.x
Level $8470$
Weight $2$
Character orbit 8470.a
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3q^{9} - q^{10} - 4q^{13} + q^{14} + q^{16} + 4q^{17} - 3q^{18} + 4q^{19} - q^{20} - 2q^{23} + q^{25} - 4q^{26} + q^{28} - 4q^{29} + 2q^{31} + q^{32} + 4q^{34} - q^{35} - 3q^{36} + 6q^{37} + 4q^{38} - q^{40} - 10q^{41} - 4q^{43} + 3q^{45} - 2q^{46} + q^{49} + q^{50} - 4q^{52} - 10q^{53} + q^{56} - 4q^{58} + 6q^{59} - 2q^{61} + 2q^{62} - 3q^{63} + q^{64} + 4q^{65} + 14q^{67} + 4q^{68} - q^{70} - 12q^{71} - 3q^{72} + 6q^{74} + 4q^{76} + 8q^{79} - q^{80} + 9q^{81} - 10q^{82} - 16q^{83} - 4q^{85} - 4q^{86} - 14q^{89} + 3q^{90} - 4q^{91} - 2q^{92} - 4q^{95} - 14q^{97} + q^{98} + O(q^{100}) \)

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 0 1.00000 −1.00000 0 1.00000 1.00000 −3.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

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 8470.2.a.x yes 1
11.b odd 2 1 8470.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.i 1 11.b odd 2 1
8470.2.a.x yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3} \)
\( T_{13} + 4 \)
\( T_{17} - 4 \)
\( T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( 4 + T \)
$17$ \( -4 + T \)
$19$ \( -4 + T \)
$23$ \( 2 + T \)
$29$ \( 4 + T \)
$31$ \( -2 + T \)
$37$ \( -6 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( -6 + T \)
$61$ \( 2 + T \)
$67$ \( -14 + T \)
$71$ \( 12 + T \)
$73$ \( T \)
$79$ \( -8 + T \)
$83$ \( 16 + T \)
$89$ \( 14 + T \)
$97$ \( 14 + T \)
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