Properties

Label 858.2.a.g
Level $858$
Weight $2$
Character orbit 858.a
Self dual yes
Analytic conductor $6.851$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 858 = 2 \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 858.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.85116449343\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} - q^{10} + q^{11} - q^{12} - q^{13} - 3q^{14} + q^{15} + q^{16} - 4q^{17} + q^{18} - 2q^{19} - q^{20} + 3q^{21} + q^{22} - q^{23} - q^{24} - 4q^{25} - q^{26} - q^{27} - 3q^{28} - 9q^{29} + q^{30} - 4q^{31} + q^{32} - q^{33} - 4q^{34} + 3q^{35} + q^{36} - 6q^{37} - 2q^{38} + q^{39} - q^{40} + q^{41} + 3q^{42} + 11q^{43} + q^{44} - q^{45} - q^{46} - q^{48} + 2q^{49} - 4q^{50} + 4q^{51} - q^{52} - 10q^{53} - q^{54} - q^{55} - 3q^{56} + 2q^{57} - 9q^{58} - 3q^{59} + q^{60} + 5q^{61} - 4q^{62} - 3q^{63} + q^{64} + q^{65} - q^{66} + 3q^{67} - 4q^{68} + q^{69} + 3q^{70} + 10q^{71} + q^{72} + 9q^{73} - 6q^{74} + 4q^{75} - 2q^{76} - 3q^{77} + q^{78} + 10q^{79} - q^{80} + q^{81} + q^{82} - 6q^{83} + 3q^{84} + 4q^{85} + 11q^{86} + 9q^{87} + q^{88} - 8q^{89} - q^{90} + 3q^{91} - q^{92} + 4q^{93} + 2q^{95} - q^{96} + 2q^{97} + 2q^{98} + q^{99} + O(q^{100}) \)

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 −3.00000 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(-1\)
\(13\) \(1\)

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 858.2.a.g 1
3.b odd 2 1 2574.2.a.i 1
4.b odd 2 1 6864.2.a.u 1
11.b odd 2 1 9438.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.g 1 1.a even 1 1 trivial
2574.2.a.i 1 3.b odd 2 1
6864.2.a.u 1 4.b odd 2 1
9438.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(858))\):

\( T_{5} + 1 \)
\( T_{7} + 3 \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( 3 + T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( 4 + T \)
$19$ \( 2 + T \)
$23$ \( 1 + T \)
$29$ \( 9 + T \)
$31$ \( 4 + T \)
$37$ \( 6 + T \)
$41$ \( -1 + T \)
$43$ \( -11 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 3 + T \)
$61$ \( -5 + T \)
$67$ \( -3 + T \)
$71$ \( -10 + T \)
$73$ \( -9 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( 8 + T \)
$97$ \( -2 + T \)
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