Properties

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

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Newspace parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 168.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

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

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

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

Character values

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

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(1 + \beta_{2}\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i −3.62132 2.09077i 2.44949i 1.62132 2.09077i 2.82843 1.50000 2.59808i 5.12132 2.95680i
5.2 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0.621320 + 0.358719i 2.44949i −2.62132 + 0.358719i −2.82843 1.50000 2.59808i 0.878680 0.507306i
101.1 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −3.62132 + 2.09077i 2.44949i 1.62132 + 2.09077i 2.82843 1.50000 + 2.59808i 5.12132 + 2.95680i
101.2 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0.621320 0.358719i 2.44949i −2.62132 0.358719i −2.82843 1.50000 + 2.59808i 0.878680 + 0.507306i
\(n\): 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 \(\Q(\sqrt{-6}) \)
7.d odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.ba.b yes 4
3.b odd 2 1 168.2.ba.a 4
4.b odd 2 1 672.2.bi.a 4
7.d odd 6 1 inner 168.2.ba.b yes 4
8.b even 2 1 168.2.ba.a 4
8.d odd 2 1 672.2.bi.b 4
12.b even 2 1 672.2.bi.b 4
21.g even 6 1 168.2.ba.a 4
24.f even 2 1 672.2.bi.a 4
24.h odd 2 1 CM 168.2.ba.b yes 4
28.f even 6 1 672.2.bi.a 4
56.j odd 6 1 168.2.ba.a 4
56.m even 6 1 672.2.bi.b 4
84.j odd 6 1 672.2.bi.b 4
168.ba even 6 1 inner 168.2.ba.b yes 4
168.be odd 6 1 672.2.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.a 4 3.b odd 2 1
168.2.ba.a 4 8.b even 2 1
168.2.ba.a 4 21.g even 6 1
168.2.ba.a 4 56.j odd 6 1
168.2.ba.b yes 4 1.a even 1 1 trivial
168.2.ba.b yes 4 7.d odd 6 1 inner
168.2.ba.b yes 4 24.h odd 2 1 CM
168.2.ba.b yes 4 168.ba even 6 1 inner
672.2.bi.a 4 4.b odd 2 1
672.2.bi.a 4 24.f even 2 1
672.2.bi.a 4 28.f even 6 1
672.2.bi.a 4 168.be odd 6 1
672.2.bi.b 4 8.d odd 2 1
672.2.bi.b 4 12.b even 2 1
672.2.bi.b 4 56.m even 6 1
672.2.bi.b 4 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6 T_{5}^{3} + 9 T_{5}^{2} - 18 T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 + 2 T^{2} + 25 T^{4} )( 1 + 6 T + 17 T^{2} + 30 T^{3} + 25 T^{4} ) \)
$7$ \( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} )( 1 - 10 T^{2} + 121 T^{4} ) \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 19 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 18 T + 137 T^{2} - 522 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 10 T + 31 T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} ) \)
$37$ \( ( 1 + 37 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 - 47 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 6 T + 65 T^{2} - 318 T^{3} + 2809 T^{4} )( 1 - 94 T^{2} + 2809 T^{4} ) \)
$59$ \( ( 1 - 10 T^{2} + 3481 T^{4} )( 1 + 18 T + 167 T^{2} + 1062 T^{3} + 3481 T^{4} ) \)
$61$ \( ( 1 - 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 67 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} )( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} ) \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} ) \)
$83$ \( ( 1 - 30 T + 383 T^{2} - 2490 T^{3} + 6889 T^{4} )( 1 + 30 T + 383 T^{2} + 2490 T^{3} + 6889 T^{4} ) \)
$89$ \( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} )( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} ) \)
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