Properties

Label 168.2.i.d
Level 168
Weight 2
Character orbit 168.i
Analytic conductor 1.341
Analytic rank 0
Dimension 8
CM discriminant -56
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: \(8\)
Coefficient field: 8.0.10070523904.11
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} + ( \beta_{6} - \beta_{7} ) q^{5} -\beta_{7} q^{6} -\beta_{3} q^{7} + 2 \beta_{2} q^{8} + ( \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} + ( \beta_{6} - \beta_{7} ) q^{5} -\beta_{7} q^{6} -\beta_{3} q^{7} + 2 \beta_{2} q^{8} + ( \beta_{2} + \beta_{3} ) q^{9} + ( \beta_{1} + \beta_{5} ) q^{10} + 2 \beta_{1} q^{12} + ( -2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{13} + \beta_{4} q^{14} + ( -1 - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{15} + 4 q^{16} + ( 2 - \beta_{4} ) q^{18} + ( \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( -2 \beta_{6} + 2 \beta_{7} ) q^{20} + ( \beta_{5} + \beta_{6} ) q^{21} + 2 \beta_{4} q^{23} + 2 \beta_{7} q^{24} + ( -5 - 2 \beta_{3} ) q^{25} + ( 3 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{26} + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{27} + 2 \beta_{3} q^{28} + ( -4 + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{30} -4 \beta_{2} q^{32} + ( -3 \beta_{1} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{35} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{36} + ( -3 \beta_{1} + \beta_{5} + 2 \beta_{6} ) q^{38} + ( 5 + 4 \beta_{2} + \beta_{3} - \beta_{4} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{40} + ( -\beta_{1} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{45} + 4 \beta_{3} q^{46} -4 \beta_{1} q^{48} + 7 q^{49} + ( 5 \beta_{2} + 2 \beta_{4} ) q^{50} + ( 4 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -\beta_{1} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{54} -2 \beta_{4} q^{56} + ( -1 - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{57} + ( 3 \beta_{1} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{59} + ( 2 + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{60} + ( 4 \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{61} + ( -7 - \beta_{4} ) q^{63} -8 q^{64} + ( -2 \beta_{2} - 4 \beta_{4} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{69} + ( -5 \beta_{1} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{70} + 4 \beta_{2} q^{71} + ( -4 + 2 \beta_{4} ) q^{72} + ( 5 \beta_{1} + 2 \beta_{5} + 2 \beta_{6} ) q^{75} + ( -2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{76} + ( 8 - 5 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{78} -2 \beta_{3} q^{79} + ( 4 \beta_{6} - 4 \beta_{7} ) q^{80} + ( 5 + 2 \beta_{4} ) q^{81} + ( 3 \beta_{1} - \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{83} + ( -2 \beta_{5} - 2 \beta_{6} ) q^{84} + ( 5 \beta_{1} + \beta_{5} + 4 \beta_{6} ) q^{90} + ( \beta_{1} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} -4 \beta_{4} q^{92} + ( -8 \beta_{2} + 2 \beta_{4} ) q^{95} -4 \beta_{7} q^{96} -7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{4} + O(q^{10}) \) \( 8q - 16q^{4} - 8q^{15} + 32q^{16} + 16q^{18} - 40q^{25} - 32q^{30} + 40q^{39} + 56q^{49} - 8q^{57} + 16q^{60} - 56q^{63} - 64q^{64} - 32q^{72} + 64q^{78} + 40q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 9 \nu^{5} + 37 \nu^{3} + 63 \nu \)\()/54\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} + 9 \nu^{5} + 20 \nu^{3} - 63 \nu \)\()/54\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{3} \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{5}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 5\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{3} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-19 \beta_{7} + \beta_{6} + \beta_{5}\)

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
1.68014 + 0.420861i
0.420861 + 1.68014i
−0.420861 1.68014i
−1.68014 0.420861i
1.68014 0.420861i
0.420861 1.68014i
−0.420861 + 1.68014i
−1.68014 + 0.420861i
1.41421i −1.68014 0.420861i −2.00000 3.91044i −0.595188 + 2.37608i −2.64575 2.82843i 2.64575 + 1.41421i 5.53019
125.2 1.41421i −0.420861 1.68014i −2.00000 2.16991i −2.37608 + 0.595188i 2.64575 2.82843i −2.64575 + 1.41421i −3.06871
125.3 1.41421i 0.420861 + 1.68014i −2.00000 2.16991i 2.37608 0.595188i 2.64575 2.82843i −2.64575 + 1.41421i 3.06871
125.4 1.41421i 1.68014 + 0.420861i −2.00000 3.91044i 0.595188 2.37608i −2.64575 2.82843i 2.64575 + 1.41421i −5.53019
125.5 1.41421i −1.68014 + 0.420861i −2.00000 3.91044i −0.595188 2.37608i −2.64575 2.82843i 2.64575 1.41421i 5.53019
125.6 1.41421i −0.420861 + 1.68014i −2.00000 2.16991i −2.37608 0.595188i 2.64575 2.82843i −2.64575 1.41421i −3.06871
125.7 1.41421i 0.420861 1.68014i −2.00000 2.16991i 2.37608 + 0.595188i 2.64575 2.82843i −2.64575 1.41421i 3.06871
125.8 1.41421i 1.68014 0.420861i −2.00000 3.91044i 0.595188 + 2.37608i −2.64575 2.82843i 2.64575 1.41421i −5.53019
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.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.d 8
3.b odd 2 1 inner 168.2.i.d 8
4.b odd 2 1 672.2.i.e 8
7.b odd 2 1 inner 168.2.i.d 8
8.b even 2 1 inner 168.2.i.d 8
8.d odd 2 1 672.2.i.e 8
12.b even 2 1 672.2.i.e 8
21.c even 2 1 inner 168.2.i.d 8
24.f even 2 1 672.2.i.e 8
24.h odd 2 1 inner 168.2.i.d 8
28.d even 2 1 672.2.i.e 8
56.e even 2 1 672.2.i.e 8
56.h odd 2 1 CM 168.2.i.d 8
84.h odd 2 1 672.2.i.e 8
168.e odd 2 1 672.2.i.e 8
168.i even 2 1 inner 168.2.i.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.d 8 1.a even 1 1 trivial
168.2.i.d 8 3.b odd 2 1 inner
168.2.i.d 8 7.b odd 2 1 inner
168.2.i.d 8 8.b even 2 1 inner
168.2.i.d 8 21.c even 2 1 inner
168.2.i.d 8 24.h odd 2 1 inner
168.2.i.d 8 56.h odd 2 1 CM
168.2.i.d 8 168.i even 2 1 inner
672.2.i.e 8 4.b odd 2 1
672.2.i.e 8 8.d odd 2 1
672.2.i.e 8 12.b even 2 1
672.2.i.e 8 24.f even 2 1
672.2.i.e 8 28.d even 2 1
672.2.i.e 8 56.e even 2 1
672.2.i.e 8 84.h odd 2 1
672.2.i.e 8 168.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 \(S_{2}^{\mathrm{new}}(168, [\chi])\):

\( T_{5}^{4} + 20 T_{5}^{2} + 72 \)
\( T_{11} \)
\( T_{13}^{4} - 52 T_{13}^{2} + 648 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{4} \)
$3$ \( 1 - 10 T^{4} + 81 T^{8} \)
$5$ \( ( 1 + 22 T^{4} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{8} \)
$13$ \( ( 1 + 310 T^{4} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 - 650 T^{4} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 6 T + 23 T^{2} )^{4}( 1 + 6 T + 23 T^{2} )^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 + 41 T^{2} )^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 + 53 T^{2} )^{8} \)
$59$ \( ( 1 - 1130 T^{4} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 7370 T^{4} + 13845841 T^{8} )^{2} \)
$67$ \( ( 1 - 67 T^{2} )^{8} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 + 130 T^{2} + 6241 T^{4} )^{4} \)
$83$ \( ( 1 - 13130 T^{4} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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