Properties

Label 168.2.a.a
Level $168$
Weight $2$
Character orbit 168.a
Self dual yes
Analytic conductor $1.341$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(1.34148675396\)
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

\(f(q)\) \(=\) \( q - q^{3} + 2 q^{5} + q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 2 q^{5} + q^{7} + q^{9} + 6 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} - q^{21} - 4 q^{23} - q^{25} - q^{27} - 10 q^{29} - 8 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} + 2 q^{45} + 8 q^{47} + q^{49} + 2 q^{51} - 10 q^{53} - 4 q^{57} + 12 q^{59} - 2 q^{61} + q^{63} + 12 q^{65} + 12 q^{67} + 4 q^{69} - 12 q^{71} - 14 q^{73} + q^{75} - 8 q^{79} + q^{81} + 12 q^{83} - 4 q^{85} + 10 q^{87} - 2 q^{89} + 6 q^{91} + 8 q^{93} + 8 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −1.00000 0 2.00000 0 1.00000 0 1.00000 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 168.2.a.a 1
3.b odd 2 1 504.2.a.e 1
4.b odd 2 1 336.2.a.e 1
5.b even 2 1 4200.2.a.t 1
5.c odd 4 2 4200.2.t.j 2
7.b odd 2 1 1176.2.a.f 1
7.c even 3 2 1176.2.q.f 2
7.d odd 6 2 1176.2.q.d 2
8.b even 2 1 1344.2.a.m 1
8.d odd 2 1 1344.2.a.b 1
12.b even 2 1 1008.2.a.b 1
16.e even 4 2 5376.2.c.d 2
16.f odd 4 2 5376.2.c.bb 2
20.d odd 2 1 8400.2.a.y 1
21.c even 2 1 3528.2.a.v 1
21.g even 6 2 3528.2.s.g 2
21.h odd 6 2 3528.2.s.w 2
24.f even 2 1 4032.2.a.bc 1
24.h odd 2 1 4032.2.a.bh 1
28.d even 2 1 2352.2.a.c 1
28.f even 6 2 2352.2.q.w 2
28.g odd 6 2 2352.2.q.d 2
56.e even 2 1 9408.2.a.da 1
56.h odd 2 1 9408.2.a.be 1
84.h odd 2 1 7056.2.a.bq 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 1.a even 1 1 trivial
336.2.a.e 1 4.b odd 2 1
504.2.a.e 1 3.b odd 2 1
1008.2.a.b 1 12.b even 2 1
1176.2.a.f 1 7.b odd 2 1
1176.2.q.d 2 7.d odd 6 2
1176.2.q.f 2 7.c even 3 2
1344.2.a.b 1 8.d odd 2 1
1344.2.a.m 1 8.b even 2 1
2352.2.a.c 1 28.d even 2 1
2352.2.q.d 2 28.g odd 6 2
2352.2.q.w 2 28.f even 6 2
3528.2.a.v 1 21.c even 2 1
3528.2.s.g 2 21.g even 6 2
3528.2.s.w 2 21.h odd 6 2
4032.2.a.bc 1 24.f even 2 1
4032.2.a.bh 1 24.h odd 2 1
4200.2.a.t 1 5.b even 2 1
4200.2.t.j 2 5.c odd 4 2
5376.2.c.d 2 16.e even 4 2
5376.2.c.bb 2 16.f odd 4 2
7056.2.a.bq 1 84.h odd 2 1
8400.2.a.y 1 20.d odd 2 1
9408.2.a.be 1 56.h odd 2 1
9408.2.a.da 1 56.e even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 6 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(168))\). Copy content Toggle raw display

Hecke characteristic polynomials

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