Properties

Label 158.2.a.d
Level $158$
Weight $2$
Character orbit 158.a
Self dual yes
Analytic conductor $1.262$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 158 = 2 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 158.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.26163635194\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3q^{7} + q^{8} - 2q^{9} + q^{10} + 2q^{11} - q^{12} - q^{13} + 3q^{14} - q^{15} + q^{16} - 2q^{17} - 2q^{18} + q^{20} - 3q^{21} + 2q^{22} - 6q^{23} - q^{24} - 4q^{25} - q^{26} + 5q^{27} + 3q^{28} - 10q^{29} - q^{30} + 2q^{31} + q^{32} - 2q^{33} - 2q^{34} + 3q^{35} - 2q^{36} - 2q^{37} + q^{39} + q^{40} + 2q^{41} - 3q^{42} + 4q^{43} + 2q^{44} - 2q^{45} - 6q^{46} + 3q^{47} - q^{48} + 2q^{49} - 4q^{50} + 2q^{51} - q^{52} + 4q^{53} + 5q^{54} + 2q^{55} + 3q^{56} - 10q^{58} + 5q^{59} - q^{60} + 12q^{61} + 2q^{62} - 6q^{63} + q^{64} - q^{65} - 2q^{66} + 8q^{67} - 2q^{68} + 6q^{69} + 3q^{70} - 13q^{71} - 2q^{72} - 6q^{73} - 2q^{74} + 4q^{75} + 6q^{77} + q^{78} - q^{79} + q^{80} + q^{81} + 2q^{82} - 6q^{83} - 3q^{84} - 2q^{85} + 4q^{86} + 10q^{87} + 2q^{88} - 15q^{89} - 2q^{90} - 3q^{91} - 6q^{92} - 2q^{93} + 3q^{94} - q^{96} + 13q^{97} + 2q^{98} - 4q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(79\) \(1\)

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 158.2.a.d 1
3.b odd 2 1 1422.2.a.a 1
4.b odd 2 1 1264.2.a.f 1
5.b even 2 1 3950.2.a.d 1
7.b odd 2 1 7742.2.a.m 1
8.b even 2 1 5056.2.a.m 1
8.d odd 2 1 5056.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
158.2.a.d 1 1.a even 1 1 trivial
1264.2.a.f 1 4.b odd 2 1
1422.2.a.a 1 3.b odd 2 1
3950.2.a.d 1 5.b even 2 1
5056.2.a.e 1 8.d odd 2 1
5056.2.a.m 1 8.b even 2 1
7742.2.a.m 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(158))\):

\( T_{3} + 1 \)
\( T_{5} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( -3 + T \)
$11$ \( -2 + T \)
$13$ \( 1 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( 6 + T \)
$29$ \( 10 + T \)
$31$ \( -2 + T \)
$37$ \( 2 + T \)
$41$ \( -2 + T \)
$43$ \( -4 + T \)
$47$ \( -3 + T \)
$53$ \( -4 + T \)
$59$ \( -5 + T \)
$61$ \( -12 + T \)
$67$ \( -8 + T \)
$71$ \( 13 + T \)
$73$ \( 6 + T \)
$79$ \( 1 + T \)
$83$ \( 6 + T \)
$89$ \( 15 + T \)
$97$ \( -13 + T \)
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