Properties

Label 168.2.i.c
Level 168
Weight 2
Character orbit 168.i
Analytic conductor 1.341
Analytic rank 0
Dimension 4
CM no
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.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: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 + \beta_{3} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{6} + ( -2 + \beta_{3} ) q^{7} + 2 \beta_{2} q^{8} + ( -1 - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{2} ) q^{3} + ( 1 + \beta_{3} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{3} ) q^{6} + ( -2 + \beta_{3} ) q^{7} + 2 \beta_{2} q^{8} + ( -1 - 2 \beta_{2} ) q^{9} + ( 1 - \beta_{3} ) q^{10} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{12} -2 q^{13} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{14} + ( -2 - \beta_{2} ) q^{15} + ( -2 + 2 \beta_{3} ) q^{16} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{17} + ( 2 - \beta_{1} - 2 \beta_{3} ) q^{18} + 4 q^{19} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{20} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( 3 + \beta_{3} ) q^{22} + \beta_{2} q^{23} + ( 4 + 2 \beta_{2} ) q^{24} + 3 q^{25} -2 \beta_{1} q^{26} + ( -5 - \beta_{2} ) q^{27} + ( -5 - \beta_{3} ) q^{28} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{30} + 4 \beta_{3} q^{31} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{32} + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{33} + ( -9 - 3 \beta_{3} ) q^{34} + ( 2 \beta_{1} + \beta_{2} ) q^{35} + ( -1 + 4 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{36} -6 \beta_{3} q^{37} + 4 \beta_{1} q^{38} + ( -2 + 2 \beta_{2} ) q^{39} + 4 q^{40} + ( 2 \beta_{1} - \beta_{2} ) q^{41} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{42} -2 \beta_{3} q^{43} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( -4 + \beta_{2} ) q^{45} + ( -1 + \beta_{3} ) q^{46} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{47} + ( -2 + 4 \beta_{1} + 2 \beta_{3} ) q^{48} + ( 1 - 4 \beta_{3} ) q^{49} + 3 \beta_{1} q^{50} + ( -6 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{51} + ( -2 - 2 \beta_{3} ) q^{52} + ( 1 - 5 \beta_{1} - \beta_{3} ) q^{54} -2 \beta_{3} q^{55} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 - 4 \beta_{2} ) q^{57} + ( -6 - 2 \beta_{3} ) q^{58} -4 \beta_{2} q^{59} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{60} + 10 q^{61} + ( -4 \beta_{1} + 8 \beta_{2} ) q^{62} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{63} -8 q^{64} + 2 \beta_{2} q^{65} + ( 3 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{66} + 2 \beta_{3} q^{67} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{68} + ( 2 + \beta_{2} ) q^{69} + ( 1 + 3 \beta_{3} ) q^{70} + \beta_{2} q^{71} + ( 8 - 2 \beta_{2} ) q^{72} + ( 6 \beta_{1} - 12 \beta_{2} ) q^{74} + ( 3 - 3 \beta_{2} ) q^{75} + ( 4 + 4 \beta_{3} ) q^{76} + ( -4 \beta_{1} + 5 \beta_{2} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{78} + 8 q^{79} + 4 \beta_{1} q^{80} + ( -7 + 4 \beta_{2} ) q^{81} + ( 3 + \beta_{3} ) q^{82} + 8 \beta_{2} q^{83} + ( -5 - 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{84} + 6 \beta_{3} q^{85} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{87} + 4 \beta_{3} q^{88} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -1 - 4 \beta_{1} + \beta_{3} ) q^{90} + ( 4 - 2 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{93} + ( 6 + 2 \beta_{3} ) q^{94} -4 \beta_{2} q^{95} + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{96} + 8 \beta_{3} q^{97} + ( 5 \beta_{1} - 8 \beta_{2} ) q^{98} + ( -2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{4} + 4q^{6} - 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{4} + 4q^{6} - 8q^{7} - 4q^{9} + 4q^{10} + 4q^{12} - 8q^{13} - 8q^{15} - 8q^{16} + 8q^{18} + 16q^{19} - 8q^{21} + 12q^{22} + 16q^{24} + 12q^{25} - 20q^{27} - 20q^{28} + 4q^{30} - 36q^{34} - 4q^{36} - 8q^{39} + 16q^{40} + 4q^{42} - 16q^{45} - 4q^{46} - 8q^{48} + 4q^{49} - 8q^{52} + 4q^{54} + 16q^{57} - 24q^{58} - 8q^{60} + 40q^{61} + 8q^{63} - 32q^{64} + 12q^{66} + 8q^{69} + 4q^{70} + 32q^{72} + 12q^{75} + 16q^{76} - 8q^{78} + 32q^{79} - 28q^{81} + 12q^{82} - 20q^{84} - 4q^{90} + 16q^{91} + 24q^{94} + 16q^{96} + O(q^{100}) \)

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

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

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
−1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i 1.00000 + 1.41421i 1.00000 + 1.73205i 1.41421i −0.224745 2.43916i −2.00000 + 1.73205i 2.82843i −1.00000 + 2.82843i 1.00000 1.73205i
125.2 −1.22474 + 0.707107i 1.00000 1.41421i 1.00000 1.73205i 1.41421i −0.224745 + 2.43916i −2.00000 1.73205i 2.82843i −1.00000 2.82843i 1.00000 + 1.73205i
125.3 1.22474 0.707107i 1.00000 + 1.41421i 1.00000 1.73205i 1.41421i 2.22474 + 1.02494i −2.00000 1.73205i 2.82843i −1.00000 + 2.82843i 1.00000 + 1.73205i
125.4 1.22474 + 0.707107i 1.00000 1.41421i 1.00000 + 1.73205i 1.41421i 2.22474 1.02494i −2.00000 + 1.73205i 2.82843i −1.00000 2.82843i 1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 6 \)
\( T_{13} + 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 10 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
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