Properties

Label 158.2.a.c
Level $158$
Weight $2$
Character orbit 158.a
Self dual yes
Analytic conductor $1.262$
Analytic rank $1$
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: \(1\)
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

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

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 −3.00000 1.00000 −3.00000 −3.00000 −3.00000 1.00000 6.00000 −3.00000
\(n\): 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.c 1
3.b odd 2 1 1422.2.a.c 1
4.b odd 2 1 1264.2.a.h 1
5.b even 2 1 3950.2.a.f 1
7.b odd 2 1 7742.2.a.n 1
8.b even 2 1 5056.2.a.v 1
8.d odd 2 1 5056.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
158.2.a.c 1 1.a even 1 1 trivial
1264.2.a.h 1 4.b odd 2 1
1422.2.a.c 1 3.b odd 2 1
3950.2.a.f 1 5.b even 2 1
5056.2.a.b 1 8.d odd 2 1
5056.2.a.v 1 8.b even 2 1
7742.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(158))\):

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

Hecke characteristic polynomials

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