Properties

Label 1710.2.a.s
Level $1710$
Weight $2$
Character orbit 1710.a
Self dual yes
Analytic conductor $13.654$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1710.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.6544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{5} + 2q^{7} + q^{8} + q^{10} + 6q^{11} + 2q^{14} + q^{16} - 2q^{17} - q^{19} + q^{20} + 6q^{22} - 4q^{23} + q^{25} + 2q^{28} + 8q^{29} - 8q^{31} + q^{32} - 2q^{34} + 2q^{35} - 4q^{37} - q^{38} + q^{40} + 4q^{41} - 6q^{43} + 6q^{44} - 4q^{46} + 12q^{47} - 3q^{49} + q^{50} - 6q^{53} + 6q^{55} + 2q^{56} + 8q^{58} + 4q^{59} + 2q^{61} - 8q^{62} + q^{64} - 8q^{67} - 2q^{68} + 2q^{70} + 6q^{73} - 4q^{74} - q^{76} + 12q^{77} + 8q^{79} + q^{80} + 4q^{82} - 4q^{83} - 2q^{85} - 6q^{86} + 6q^{88} + 4q^{89} - 4q^{92} + 12q^{94} - q^{95} + 12q^{97} - 3q^{98} + O(q^{100}) \)

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 2.00000 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 1710.2.a.s 1
3.b odd 2 1 570.2.a.b 1
5.b even 2 1 8550.2.a.g 1
12.b even 2 1 4560.2.a.r 1
15.d odd 2 1 2850.2.a.y 1
15.e even 4 2 2850.2.d.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.b 1 3.b odd 2 1
1710.2.a.s 1 1.a even 1 1 trivial
2850.2.a.y 1 15.d odd 2 1
2850.2.d.j 2 15.e even 4 2
4560.2.a.r 1 12.b even 2 1
8550.2.a.g 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7} - 2 \)
\( T_{11} - 6 \)
\( T_{13} \)
\( T_{53} + 6 \)

Hecke characteristic polynomials

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