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Properties

Label 168.2.j.a
Level 168
Weight 2
Character orbit 168.j
Analytic conductor 1.341
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( 2 - 2 \beta_{2} ) q^{8} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( -\beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{6} + \beta_{2} q^{7} + ( 2 - 2 \beta_{2} ) q^{8} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} + 2 \beta_{1} q^{10} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -2 + 2 \beta_{3} ) q^{12} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{13} + ( 1 - \beta_{2} ) q^{14} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} -4 q^{16} -2 \beta_{2} q^{17} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{18} + ( 4 - \beta_{1} + \beta_{3} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( -1 + \beta_{3} ) q^{21} + ( 2 - 2 \beta_{2} - 4 \beta_{3} ) q^{22} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{24} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -4 - 3 \beta_{2} + \beta_{3} ) q^{27} -2 q^{28} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{30} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{31} + ( 4 + 4 \beta_{2} ) q^{32} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{33} + ( -2 + 2 \beta_{2} ) q^{34} + ( -\beta_{1} - \beta_{3} ) q^{35} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{36} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{38} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{39} + 4 \beta_{3} q^{40} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{42} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -4 - 4 \beta_{1} + 4 \beta_{3} ) q^{44} + ( 4 - 3 \beta_{1} - \beta_{3} ) q^{45} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{46} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{47} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{48} - q^{49} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{50} + ( 2 - 2 \beta_{3} ) q^{51} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{52} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 1 + \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{54} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{55} + ( 2 + 2 \beta_{2} ) q^{56} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{57} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( 4 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -4 + 4 \beta_{2} + 8 \beta_{3} ) q^{62} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} -8 \beta_{2} q^{64} -4 \beta_{2} q^{65} + ( 10 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( 2 + 4 \beta_{1} - 4 \beta_{3} ) q^{67} + 4 q^{68} + ( 4 + 4 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{69} + 2 \beta_{3} q^{70} + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 2 + 6 \beta_{2} + 4 \beta_{3} ) q^{72} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{73} -4 \beta_{3} q^{74} + ( -4 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{75} + ( -2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{76} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{77} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{78} + ( 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{80} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{81} + ( -2 + 2 \beta_{2} + 4 \beta_{3} ) q^{82} + ( -\beta_{1} + 10 \beta_{2} - \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{84} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{85} + ( -6 - 4 \beta_{1} - 6 \beta_{2} ) q^{86} + ( 4 + 8 \beta_{1} + 4 \beta_{2} ) q^{88} -6 \beta_{2} q^{89} + ( -4 + 2 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{90} + ( -2 - \beta_{1} + \beta_{3} ) q^{91} + ( 4 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} ) q^{92} + ( 12 + 4 \beta_{1} - 8 \beta_{2} ) q^{93} + ( 4 + 4 \beta_{1} + 4 \beta_{2} ) q^{94} + ( 4 - 6 \beta_{1} + 6 \beta_{3} ) q^{95} + ( -4 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{96} + ( 10 + 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 1 + \beta_{2} ) q^{98} + ( -4 - 4 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 2q^{3} + 4q^{5} + 4q^{6} + 8q^{8} + O(q^{10}) \) \( 4q - 4q^{2} - 2q^{3} + 4q^{5} + 4q^{6} + 8q^{8} - 4q^{10} - 4q^{12} + 4q^{14} - 12q^{15} - 16q^{16} + 8q^{18} + 20q^{19} - 2q^{21} + 16q^{23} + 4q^{25} + 4q^{26} - 14q^{27} - 8q^{28} + 4q^{30} + 16q^{32} - 20q^{33} - 8q^{34} - 16q^{36} - 20q^{38} - 12q^{39} + 8q^{40} + 16q^{43} + 20q^{45} - 16q^{46} - 8q^{47} + 8q^{48} - 4q^{49} - 4q^{50} + 4q^{51} - 8q^{52} - 24q^{53} + 8q^{56} - 20q^{57} + 16q^{60} - 8q^{63} + 40q^{66} - 8q^{67} + 16q^{68} + 12q^{69} + 4q^{70} + 8q^{71} + 16q^{72} + 16q^{73} - 8q^{74} - 22q^{75} + 20q^{78} - 16q^{80} + 4q^{81} + 4q^{84} - 16q^{86} - 12q^{90} - 4q^{91} + 40q^{93} + 8q^{94} + 40q^{95} - 16q^{96} + 32q^{97} + 4q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} + \nu + 1 \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu - 1 \)
Display \(\nu^j\) in terms of \(\beta_i\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{1} - 2\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)
Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
− 1.61803i
0.618034i
1.61803i
− 0.618034i
−1.00000 − 1.00000i −1.61803 − 0.618034i 2.00000i 3.23607 1.00000 + 2.23607i 1.00000i 2.00000 − 2.00000i 2.23607 + 2.00000i −3.23607 − 3.23607i
155.2 −1.00000 − 1.00000i 0.618034 + 1.61803i 2.00000i −1.23607 1.00000 − 2.23607i 1.00000i 2.00000 − 2.00000i −2.23607 + 2.00000i 1.23607 + 1.23607i
155.3 −1.00000 + 1.00000i −1.61803 + 0.618034i − 2.00000i 3.23607 1.00000 − 2.23607i − 1.00000i 2.00000 + 2.00000i 2.23607 − 2.00000i −3.23607 + 3.23607i
155.4 −1.00000 + 1.00000i 0.618034 − 1.61803i − 2.00000i −1.23607 1.00000 + 2.23607i − 1.00000i 2.00000 + 2.00000i −2.23607 − 2.00000i 1.23607 − 1.23607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.f Even 1 yes

Hecke kernels

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