Properties

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

Related objects

Downloads

Learn more

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: \(2\)
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

\(f(q)\) \(=\) \( q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 2q^{12} - 2q^{13} - q^{14} + 2q^{15} + q^{16} - 2q^{17} - q^{18} - 8q^{19} - q^{20} - 2q^{21} - 6q^{23} + 2q^{24} + q^{25} + 2q^{26} + 4q^{27} + q^{28} - 8q^{29} - 2q^{30} - 8q^{31} - q^{32} + 2q^{34} - q^{35} + q^{36} - 10q^{37} + 8q^{38} + 4q^{39} + q^{40} + 2q^{42} - 4q^{43} - q^{45} + 6q^{46} - 6q^{47} - 2q^{48} + q^{49} - q^{50} + 4q^{51} - 2q^{52} - 2q^{53} - 4q^{54} - q^{56} + 16q^{57} + 8q^{58} + 4q^{59} + 2q^{60} + 4q^{61} + 8q^{62} + q^{63} + q^{64} + 2q^{65} + 2q^{67} - 2q^{68} + 12q^{69} + q^{70} + 8q^{71} - q^{72} + 6q^{73} + 10q^{74} - 2q^{75} - 8q^{76} - 4q^{78} - 4q^{79} - q^{80} - 11q^{81} - 4q^{83} - 2q^{84} + 2q^{85} + 4q^{86} + 16q^{87} - 2q^{89} + q^{90} - 2q^{91} - 6q^{92} + 16q^{93} + 6q^{94} + 8q^{95} + 2q^{96} - 2q^{97} - q^{98} + O(q^{100}) \)

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 −2.00000 1.00000 −1.00000 2.00000 1.00000 −1.00000 1.00000 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.d 1
11.b odd 2 1 8470.2.a.s yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8470.2.a.d 1 1.a even 1 1 trivial
8470.2.a.s yes 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3} + 2 \)
\( T_{13} + 2 \)
\( T_{17} + 2 \)
\( T_{19} + 8 \)

Hecke characteristic polynomials

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