Properties

Label 168.4.a.c
Level $168$
Weight $4$
Character orbit 168.a
Self dual yes
Analytic conductor $9.912$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.91232088096\)
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

\(f(q)\) \(=\) \( q - 3q^{3} + 4q^{5} - 7q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 4q^{5} - 7q^{7} + 9q^{9} - 26q^{11} + 2q^{13} - 12q^{15} - 36q^{17} - 76q^{19} + 21q^{21} - 114q^{23} - 109q^{25} - 27q^{27} + 6q^{29} - 256q^{31} + 78q^{33} - 28q^{35} - 86q^{37} - 6q^{39} + 160q^{41} - 220q^{43} + 36q^{45} + 308q^{47} + 49q^{49} + 108q^{51} + 258q^{53} - 104q^{55} + 228q^{57} + 264q^{59} + 606q^{61} - 63q^{63} + 8q^{65} - 520q^{67} + 342q^{69} - 286q^{71} - 530q^{73} + 327q^{75} + 182q^{77} - 44q^{79} + 81q^{81} + 1012q^{83} - 144q^{85} - 18q^{87} + 768q^{89} - 14q^{91} + 768q^{93} - 304q^{95} + 222q^{97} - 234q^{99} + O(q^{100}) \)

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
0 −3.00000 0 4.00000 0 −7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 168.4.a.c 1
3.b odd 2 1 504.4.a.b 1
4.b odd 2 1 336.4.a.j 1
7.b odd 2 1 1176.4.a.j 1
8.b even 2 1 1344.4.a.s 1
8.d odd 2 1 1344.4.a.e 1
12.b even 2 1 1008.4.a.i 1
28.d even 2 1 2352.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.c 1 1.a even 1 1 trivial
336.4.a.j 1 4.b odd 2 1
504.4.a.b 1 3.b odd 2 1
1008.4.a.i 1 12.b even 2 1
1176.4.a.j 1 7.b odd 2 1
1344.4.a.e 1 8.d odd 2 1
1344.4.a.s 1 8.b even 2 1
2352.4.a.h 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(168))\):

\( T_{5} - 4 \)
\( T_{11} + 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( -4 + T \)
$7$ \( 7 + T \)
$11$ \( 26 + T \)
$13$ \( -2 + T \)
$17$ \( 36 + T \)
$19$ \( 76 + T \)
$23$ \( 114 + T \)
$29$ \( -6 + T \)
$31$ \( 256 + T \)
$37$ \( 86 + T \)
$41$ \( -160 + T \)
$43$ \( 220 + T \)
$47$ \( -308 + T \)
$53$ \( -258 + T \)
$59$ \( -264 + T \)
$61$ \( -606 + T \)
$67$ \( 520 + T \)
$71$ \( 286 + T \)
$73$ \( 530 + T \)
$79$ \( 44 + T \)
$83$ \( -1012 + T \)
$89$ \( -768 + T \)
$97$ \( -222 + T \)
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