Properties

Label 168.2.i.e
Level 168
Weight 2
Character orbit 168.i
Analytic conductor 1.341
Analytic rank 0
Dimension 8
CM no
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: 8.0.3317760000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{4} - \beta_{7} ) q^{6} + ( 1 - \beta_{6} + 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} - \beta_{5} ) q^{8} + ( 2 + \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -\beta_{4} - \beta_{6} ) q^{3} + ( -1 - \beta_{2} ) q^{4} + \beta_{6} q^{5} + ( \beta_{1} + \beta_{4} - \beta_{7} ) q^{6} + ( 1 - \beta_{6} + 2 \beta_{7} ) q^{7} + ( 2 \beta_{3} - \beta_{5} ) q^{8} + ( 2 + \beta_{3} + \beta_{5} ) q^{9} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{10} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{11} + ( -\beta_{1} + \beta_{6} - 2 \beta_{7} ) q^{12} + ( 2 \beta_{4} + \beta_{6} ) q^{13} + ( 2 \beta_{1} + \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{14} + ( 1 - \beta_{3} - \beta_{5} ) q^{15} + ( -3 + \beta_{2} ) q^{16} + ( -3 - \beta_{2} + 2 \beta_{5} ) q^{18} + ( 2 \beta_{4} + \beta_{6} ) q^{19} + ( 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{20} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{21} + ( -2 + 2 \beta_{2} ) q^{22} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{23} + ( -3 \beta_{1} - \beta_{4} - 4 \beta_{6} + 3 \beta_{7} ) q^{24} + 3 q^{25} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} + ( -\beta_{4} - 4 \beta_{6} ) q^{27} + ( -1 - \beta_{2} - 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{28} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{29} + ( 3 + \beta_{2} + \beta_{5} ) q^{30} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{31} + ( -2 \beta_{3} - 3 \beta_{5} ) q^{32} + ( -4 \beta_{1} - 2 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{33} + ( 2 \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{35} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{36} + ( -2 \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{38} + ( -5 - \beta_{3} - \beta_{5} ) q^{39} + ( -\beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{40} + ( 8 \beta_{1} + 4 \beta_{4} + 6 \beta_{6} - 4 \beta_{7} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{42} + ( -2 - 4 \beta_{2} ) q^{43} + ( -4 \beta_{3} - 2 \beta_{5} ) q^{44} + ( -2 \beta_{4} + \beta_{6} ) q^{45} + ( 6 + 2 \beta_{2} ) q^{46} + ( -8 \beta_{1} - 4 \beta_{4} - 6 \beta_{6} + 4 \beta_{7} ) q^{47} + ( \beta_{1} + 4 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{48} + ( -5 - 2 \beta_{6} + 4 \beta_{7} ) q^{49} + 3 \beta_{5} q^{50} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} ) q^{52} + ( \beta_{1} + 4 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{54} + ( 2 \beta_{6} - 4 \beta_{7} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{56} + ( -5 - \beta_{3} - \beta_{5} ) q^{57} + ( 4 - 4 \beta_{2} ) q^{58} + 7 \beta_{6} q^{59} + ( -1 - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{60} + ( 2 \beta_{4} + \beta_{6} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{4} - 6 \beta_{6} + 2 \beta_{7} ) q^{62} + ( 2 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{63} + ( 7 + 3 \beta_{2} ) q^{64} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{65} + ( 2 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} + 4 \beta_{7} ) q^{66} + ( 2 + 4 \beta_{2} ) q^{67} + ( -2 \beta_{4} + 4 \beta_{6} ) q^{69} + ( -2 + 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{70} + ( 4 \beta_{3} + 4 \beta_{5} ) q^{71} + ( -1 + 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} ) q^{72} + ( 6 \beta_{6} - 12 \beta_{7} ) q^{73} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{75} + ( -\beta_{4} - 3 \beta_{6} + 5 \beta_{7} ) q^{76} + ( 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} ) q^{77} + ( 3 + \beta_{2} - 5 \beta_{5} ) q^{78} -10 q^{79} + ( -2 \beta_{1} - \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{80} + ( -1 + 4 \beta_{3} + 4 \beta_{5} ) q^{81} + ( -6 \beta_{4} + 2 \beta_{6} - 10 \beta_{7} ) q^{82} -5 \beta_{6} q^{83} + ( 7 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{84} + ( 8 \beta_{3} - 2 \beta_{5} ) q^{86} + ( 8 \beta_{1} + 4 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} ) q^{87} + ( 10 + 2 \beta_{2} ) q^{88} + ( 2 \beta_{1} - \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{90} + ( -2 - 4 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{91} + ( -4 \beta_{3} + 6 \beta_{5} ) q^{92} + ( -2 - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{93} + ( 6 \beta_{4} - 2 \beta_{6} + 10 \beta_{7} ) q^{94} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{95} + ( -\beta_{1} - 3 \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{96} + ( -2 \beta_{6} + 4 \beta_{7} ) q^{97} + ( 4 \beta_{1} + 2 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{98} + ( 2 + 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{4} + 8q^{7} + 16q^{9} + O(q^{10}) \) \( 8q - 4q^{4} + 8q^{7} + 16q^{9} + 8q^{15} - 28q^{16} - 20q^{18} - 24q^{22} + 24q^{25} - 4q^{28} + 20q^{30} - 8q^{36} - 40q^{39} + 12q^{42} + 40q^{46} - 40q^{49} - 40q^{57} + 48q^{58} - 4q^{60} + 16q^{63} + 44q^{64} - 24q^{70} - 20q^{72} + 20q^{78} - 80q^{79} - 8q^{81} + 60q^{84} + 72q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 13 x^{4} + 12 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 4 \nu^{2} - 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 6 \nu^{4} - 17 \nu^{2} + 18 \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 14 \nu^{5} - 97 \nu^{3} + 138 \nu \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 6 \nu^{4} - 5 \nu^{2} - 6 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 14 \nu^{5} - 25 \nu^{3} - 78 \nu \)\()/72\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} - 22 \nu^{5} + 5 \nu^{3} + 30 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 2 \beta_{3} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 2 \beta_{4} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{5} - 8 \beta_{3} + 3 \beta_{2} + 11\)
\(\nu^{5}\)\(=\)\(3 \beta_{7} + 10 \beta_{6} - 5 \beta_{4} + 15 \beta_{1}\)
\(\nu^{6}\)\(=\)\(7 \beta_{5} - 38 \beta_{3} + 18 \beta_{2} + 50\)
\(\nu^{7}\)\(=\)\(42 \beta_{7} + 43 \beta_{6} - 20 \beta_{4} + 57 \beta_{1}\)

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
0.578737 + 0.965926i
−0.578737 0.965926i
0.578737 0.965926i
−0.578737 + 0.965926i
−2.15988 0.258819i
2.15988 + 0.258819i
−2.15988 + 0.258819i
2.15988 0.258819i
−0.866025 1.11803i −1.58114 + 0.707107i −0.500000 + 1.93649i 1.41421i 2.15988 + 1.15539i 1.00000 + 2.44949i 2.59808 1.11803i 2.00000 2.23607i −1.58114 + 1.22474i
125.2 −0.866025 1.11803i 1.58114 0.707107i −0.500000 + 1.93649i 1.41421i −2.15988 1.15539i 1.00000 2.44949i 2.59808 1.11803i 2.00000 2.23607i 1.58114 1.22474i
125.3 −0.866025 + 1.11803i −1.58114 0.707107i −0.500000 1.93649i 1.41421i 2.15988 1.15539i 1.00000 2.44949i 2.59808 + 1.11803i 2.00000 + 2.23607i −1.58114 1.22474i
125.4 −0.866025 + 1.11803i 1.58114 + 0.707107i −0.500000 1.93649i 1.41421i −2.15988 + 1.15539i 1.00000 + 2.44949i 2.59808 + 1.11803i 2.00000 + 2.23607i 1.58114 + 1.22474i
125.5 0.866025 1.11803i −1.58114 + 0.707107i −0.500000 1.93649i 1.41421i −0.578737 + 2.38014i 1.00000 2.44949i −2.59808 1.11803i 2.00000 2.23607i −1.58114 1.22474i
125.6 0.866025 1.11803i 1.58114 0.707107i −0.500000 1.93649i 1.41421i 0.578737 2.38014i 1.00000 + 2.44949i −2.59808 1.11803i 2.00000 2.23607i 1.58114 + 1.22474i
125.7 0.866025 + 1.11803i −1.58114 0.707107i −0.500000 + 1.93649i 1.41421i −0.578737 2.38014i 1.00000 + 2.44949i −2.59808 + 1.11803i 2.00000 + 2.23607i −1.58114 + 1.22474i
125.8 0.866025 + 1.11803i 1.58114 + 0.707107i −0.500000 + 1.93649i 1.41421i 0.578737 + 2.38014i 1.00000 2.44949i −2.59808 + 1.11803i 2.00000 + 2.23607i 1.58114 1.22474i
\(n\): 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
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
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.e 8
3.b odd 2 1 inner 168.2.i.e 8
4.b odd 2 1 672.2.i.d 8
7.b odd 2 1 inner 168.2.i.e 8
8.b even 2 1 inner 168.2.i.e 8
8.d odd 2 1 672.2.i.d 8
12.b even 2 1 672.2.i.d 8
21.c even 2 1 inner 168.2.i.e 8
24.f even 2 1 672.2.i.d 8
24.h odd 2 1 inner 168.2.i.e 8
28.d even 2 1 672.2.i.d 8
56.e even 2 1 672.2.i.d 8
56.h odd 2 1 inner 168.2.i.e 8
84.h odd 2 1 672.2.i.d 8
168.e odd 2 1 672.2.i.d 8
168.i even 2 1 inner 168.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.e 8 1.a even 1 1 trivial
168.2.i.e 8 3.b odd 2 1 inner
168.2.i.e 8 7.b odd 2 1 inner
168.2.i.e 8 8.b even 2 1 inner
168.2.i.e 8 21.c even 2 1 inner
168.2.i.e 8 24.h odd 2 1 inner
168.2.i.e 8 56.h odd 2 1 inner
168.2.i.e 8 168.i even 2 1 inner
672.2.i.d 8 4.b odd 2 1
672.2.i.d 8 8.d odd 2 1
672.2.i.d 8 12.b even 2 1
672.2.i.d 8 24.f even 2 1
672.2.i.d 8 28.d even 2 1
672.2.i.d 8 56.e even 2 1
672.2.i.d 8 84.h odd 2 1
672.2.i.d 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 \(S_{2}^{\mathrm{new}}(168, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ \( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{4} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{4} \)
$11$ \( ( 1 + 10 T^{2} + 121 T^{4} )^{4} \)
$13$ \( ( 1 + 16 T^{2} + 169 T^{4} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{8} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 26 T^{2} + 529 T^{4} )^{4} \)
$29$ \( ( 1 + 10 T^{2} + 841 T^{4} )^{4} \)
$31$ \( ( 1 - 10 T + 31 T^{2} )^{4}( 1 + 10 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 - 38 T^{2} + 1681 T^{4} )^{4} \)
$43$ \( ( 1 - 26 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 26 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{8} \)
$59$ \( ( 1 - 20 T^{2} + 3481 T^{4} )^{4} \)
$61$ \( ( 1 + 112 T^{2} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 74 T^{2} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 62 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 + 70 T^{2} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 + 10 T + 79 T^{2} )^{8} \)
$83$ \( ( 1 - 116 T^{2} + 6889 T^{4} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{8} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{4} \)
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