Properties

Label 510.2.a.d
Level $510$
Weight $2$
Character orbit 510.a
Self dual yes
Analytic conductor $4.072$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.07237050309\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 4q^{13} + 2q^{14} + q^{15} + q^{16} - q^{17} + q^{18} + 4q^{19} - q^{20} - 2q^{21} + 4q^{23} - q^{24} + q^{25} + 4q^{26} - q^{27} + 2q^{28} + 2q^{29} + q^{30} + q^{32} - q^{34} - 2q^{35} + q^{36} - 2q^{37} + 4q^{38} - 4q^{39} - q^{40} - 4q^{41} - 2q^{42} + 10q^{43} - q^{45} + 4q^{46} - 8q^{47} - q^{48} - 3q^{49} + q^{50} + q^{51} + 4q^{52} + 2q^{53} - q^{54} + 2q^{56} - 4q^{57} + 2q^{58} - 2q^{59} + q^{60} - 14q^{61} + 2q^{63} + q^{64} - 4q^{65} + 2q^{67} - q^{68} - 4q^{69} - 2q^{70} - 6q^{71} + q^{72} - 4q^{73} - 2q^{74} - q^{75} + 4q^{76} - 4q^{78} - 12q^{79} - q^{80} + q^{81} - 4q^{82} + 8q^{83} - 2q^{84} + q^{85} + 10q^{86} - 2q^{87} - 10q^{89} - q^{90} + 8q^{91} + 4q^{92} - 8q^{94} - 4q^{95} - q^{96} + 8q^{97} - 3q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(17\) \(1\)

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 510.2.a.d 1
3.b odd 2 1 1530.2.a.h 1
4.b odd 2 1 4080.2.a.r 1
5.b even 2 1 2550.2.a.i 1
5.c odd 4 2 2550.2.d.f 2
15.d odd 2 1 7650.2.a.bn 1
17.b even 2 1 8670.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.d 1 1.a even 1 1 trivial
1530.2.a.h 1 3.b odd 2 1
2550.2.a.i 1 5.b even 2 1
2550.2.d.f 2 5.c odd 4 2
4080.2.a.r 1 4.b odd 2 1
7650.2.a.bn 1 15.d odd 2 1
8670.2.a.y 1 17.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(510))\):

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

Hecke characteristic polynomials

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