Properties

Label 1710.2.a.f
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

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{5} + 4 q^{7} - q^{8} + q^{10} + 4 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} + 2 q^{17} - q^{19} - q^{20} - 4 q^{22} + 8 q^{23} + q^{25} + 2 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 2 q^{34} - 4 q^{35} - 10 q^{37} + q^{38} + q^{40} + 2 q^{41} + 12 q^{43} + 4 q^{44} - 8 q^{46} + 9 q^{49} - q^{50} - 2 q^{52} - 6 q^{53} - 4 q^{55} - 4 q^{56} + 6 q^{58} - 10 q^{61} - 4 q^{62} + q^{64} + 2 q^{65} - 4 q^{67} + 2 q^{68} + 4 q^{70} + 8 q^{71} + 2 q^{73} + 10 q^{74} - q^{76} + 16 q^{77} - 12 q^{79} - q^{80} - 2 q^{82} + 8 q^{83} - 2 q^{85} - 12 q^{86} - 4 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{92} + q^{95} + 18 q^{97} - 9 q^{98} + O(q^{100}) \)

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 4.00000 −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.f 1
3.b odd 2 1 570.2.a.m 1
5.b even 2 1 8550.2.a.t 1
12.b even 2 1 4560.2.a.k 1
15.d odd 2 1 2850.2.a.a 1
15.e even 4 2 2850.2.d.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.m 1 3.b odd 2 1
1710.2.a.f 1 1.a even 1 1 trivial
2850.2.a.a 1 15.d odd 2 1
2850.2.d.k 2 15.e even 4 2
4560.2.a.k 1 12.b even 2 1
8550.2.a.t 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1710))\):

\( T_{7} - 4 \)
\( T_{11} - 4 \)
\( T_{13} + 2 \)
\( T_{53} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -4 + T \)
$11$ \( -4 + T \)
$13$ \( 2 + T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( -8 + T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( 10 + T \)
$41$ \( -2 + T \)
$43$ \( -12 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( 4 + T \)
$71$ \( -8 + T \)
$73$ \( -2 + T \)
$79$ \( 12 + T \)
$83$ \( -8 + T \)
$89$ \( 6 + T \)
$97$ \( -18 + T \)
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