Properties

Label 168.2.i.b
Level 168
Weight 2
Character orbit 168.i
Analytic conductor 1.341
Analytic rank 0
Dimension 4
CM discriminant -24
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 168.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + 2 \beta_{2} q^{5} + \beta_{3} q^{6} + ( 1 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + 2 \beta_{2} q^{5} + \beta_{3} q^{6} + ( 1 + \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} -2 \beta_{3} q^{10} -4 \beta_{1} q^{11} -2 \beta_{2} q^{12} + ( \beta_{1} - 2 \beta_{2} ) q^{14} + 6 q^{15} + 4 q^{16} -3 \beta_{1} q^{18} + 4 \beta_{2} q^{20} + ( -3 \beta_{1} - \beta_{2} ) q^{21} -8 q^{22} + 2 \beta_{3} q^{24} -7 q^{25} + 3 \beta_{2} q^{27} + ( 2 + 2 \beta_{3} ) q^{28} -2 \beta_{1} q^{29} + 6 \beta_{1} q^{30} -2 \beta_{3} q^{31} + 4 \beta_{1} q^{32} -4 \beta_{3} q^{33} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{35} -6 q^{36} -4 \beta_{3} q^{40} + ( -6 + \beta_{3} ) q^{42} -8 \beta_{1} q^{44} -6 \beta_{2} q^{45} -4 \beta_{2} q^{48} + ( -5 + 2 \beta_{3} ) q^{49} -7 \beta_{1} q^{50} + 10 \beta_{1} q^{53} -3 \beta_{3} q^{54} + 8 \beta_{3} q^{55} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{56} -4 q^{58} -6 \beta_{2} q^{59} + 12 q^{60} + 4 \beta_{2} q^{62} + ( -3 - 3 \beta_{3} ) q^{63} + 8 q^{64} + 8 \beta_{2} q^{66} + ( 12 - 2 \beta_{3} ) q^{70} -6 \beta_{1} q^{72} + 4 \beta_{3} q^{73} + 7 \beta_{2} q^{75} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{77} + 10 q^{79} + 8 \beta_{2} q^{80} + 9 q^{81} -10 \beta_{2} q^{83} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{84} -2 \beta_{3} q^{87} -16 q^{88} + 6 \beta_{3} q^{90} + 6 \beta_{1} q^{93} + 4 \beta_{3} q^{96} -8 \beta_{3} q^{97} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{98} + 12 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 4q^{7} - 12q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 4q^{7} - 12q^{9} + 24q^{15} + 16q^{16} - 32q^{22} - 28q^{25} + 8q^{28} - 24q^{36} - 24q^{42} - 20q^{49} - 16q^{58} + 48q^{60} - 12q^{63} + 32q^{64} + 48q^{70} + 40q^{79} + 36q^{81} - 64q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 + 2.44949i −2.82843 −3.00000 4.89898i
125.2 −1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 2.44949i −2.82843 −3.00000 4.89898i
125.3 1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 2.44949i 2.82843 −3.00000 4.89898i
125.4 1.41421 1.73205i 2.00000 3.46410i 2.44949i 1.00000 + 2.44949i 2.82843 −3.00000 4.89898i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.i.b 4
3.b odd 2 1 inner 168.2.i.b 4
4.b odd 2 1 672.2.i.b 4
7.b odd 2 1 inner 168.2.i.b 4
8.b even 2 1 inner 168.2.i.b 4
8.d odd 2 1 672.2.i.b 4
12.b even 2 1 672.2.i.b 4
21.c even 2 1 inner 168.2.i.b 4
24.f even 2 1 672.2.i.b 4
24.h odd 2 1 CM 168.2.i.b 4
28.d even 2 1 672.2.i.b 4
56.e even 2 1 672.2.i.b 4
56.h odd 2 1 inner 168.2.i.b 4
84.h odd 2 1 672.2.i.b 4
168.e odd 2 1 672.2.i.b 4
168.i even 2 1 inner 168.2.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.b 4 1.a even 1 1 trivial
168.2.i.b 4 3.b odd 2 1 inner
168.2.i.b 4 7.b odd 2 1 inner
168.2.i.b 4 8.b even 2 1 inner
168.2.i.b 4 21.c even 2 1 inner
168.2.i.b 4 24.h odd 2 1 CM
168.2.i.b 4 56.h odd 2 1 inner
168.2.i.b 4 168.i even 2 1 inner
672.2.i.b 4 4.b odd 2 1
672.2.i.b 4 8.d odd 2 1
672.2.i.b 4 12.b even 2 1
672.2.i.b 4 24.f even 2 1
672.2.i.b 4 28.d even 2 1
672.2.i.b 4 56.e even 2 1
672.2.i.b 4 84.h odd 2 1
672.2.i.b 4 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{11}^{2} - 32 \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( ( 1 + 3 T^{2} )^{2} \)
$5$ \( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 10 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{2}( 1 + 10 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 94 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 134 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 2 T + 97 T^{2} )^{2}( 1 + 2 T + 97 T^{2} )^{2} \)
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