Properties

Label 8470.2.a.j
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: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{5} + q^{7} - q^{8} - 3q^{9} + q^{10} + 6q^{13} - q^{14} + q^{16} - 2q^{17} + 3q^{18} - q^{20} + q^{25} - 6q^{26} + q^{28} - 6q^{29} + 8q^{31} - q^{32} + 2q^{34} - q^{35} - 3q^{36} - 10q^{37} + q^{40} - 2q^{41} - 4q^{43} + 3q^{45} + 8q^{47} + q^{49} - q^{50} + 6q^{52} - 2q^{53} - q^{56} + 6q^{58} - 8q^{59} + 14q^{61} - 8q^{62} - 3q^{63} + q^{64} - 6q^{65} - 12q^{67} - 2q^{68} + q^{70} - 16q^{71} + 3q^{72} - 2q^{73} + 10q^{74} + 8q^{79} - q^{80} + 9q^{81} + 2q^{82} - 8q^{83} + 2q^{85} + 4q^{86} + 10q^{89} - 3q^{90} + 6q^{91} - 8q^{94} + 2q^{97} - q^{98} + O(q^{100}) \)

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 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.j 1
11.b odd 2 1 70.2.a.a 1
33.d even 2 1 630.2.a.d 1
44.c even 2 1 560.2.a.d 1
55.d odd 2 1 350.2.a.b 1
55.e even 4 2 350.2.c.b 2
77.b even 2 1 490.2.a.h 1
77.h odd 6 2 490.2.e.d 2
77.i even 6 2 490.2.e.c 2
88.b odd 2 1 2240.2.a.n 1
88.g even 2 1 2240.2.a.q 1
132.d odd 2 1 5040.2.a.bm 1
165.d even 2 1 3150.2.a.bj 1
165.l odd 4 2 3150.2.g.c 2
220.g even 2 1 2800.2.a.m 1
220.i odd 4 2 2800.2.g.n 2
231.h odd 2 1 4410.2.a.b 1
308.g odd 2 1 3920.2.a.t 1
385.h even 2 1 2450.2.a.l 1
385.l odd 4 2 2450.2.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 11.b odd 2 1
350.2.a.b 1 55.d odd 2 1
350.2.c.b 2 55.e even 4 2
490.2.a.h 1 77.b even 2 1
490.2.e.c 2 77.i even 6 2
490.2.e.d 2 77.h odd 6 2
560.2.a.d 1 44.c even 2 1
630.2.a.d 1 33.d even 2 1
2240.2.a.n 1 88.b odd 2 1
2240.2.a.q 1 88.g even 2 1
2450.2.a.l 1 385.h even 2 1
2450.2.c.k 2 385.l odd 4 2
2800.2.a.m 1 220.g even 2 1
2800.2.g.n 2 220.i odd 4 2
3150.2.a.bj 1 165.d even 2 1
3150.2.g.c 2 165.l odd 4 2
3920.2.a.t 1 308.g odd 2 1
4410.2.a.b 1 231.h odd 2 1
5040.2.a.bm 1 132.d odd 2 1
8470.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):

\( T_{3} \)
\( T_{13} - 6 \)
\( T_{17} + 2 \)
\( T_{19} \)

Hecke characteristic polynomials

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