Properties

Label 336.2.bq.a
Level 336
Weight 2
Character orbit 336.bq
Analytic conductor 2.683
Analytic rank 0
Dimension 8
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bq (of order \(12\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + \zeta_{24}^{5} q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( 1 + \zeta_{24}^{6} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + \zeta_{24}^{5} q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{5} + ( 1 + \zeta_{24}^{6} ) q^{6} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} ) q^{9} + ( -1 + \zeta_{24}^{2} + \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{10} + ( 1 + \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{11} + 2 \zeta_{24} q^{12} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{13} + ( -2 + \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{14} + ( -1 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{15} -4 \zeta_{24}^{4} q^{16} + ( 4 + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{17} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{18} + ( 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{19} + ( 4 + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{6} ) q^{20} + ( -2 - \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{21} + ( 1 + 2 \zeta_{24} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{22} + ( -3 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{23} + ( 2 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} ) q^{24} + ( -4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{25} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{3} q^{27} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{28} + ( 4 + 3 \zeta_{24} - 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{29} + ( -\zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{30} + ( 1 - \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{32} + ( \zeta_{24} + \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{33} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{34} + ( -1 - 5 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{35} + 2 \zeta_{24}^{6} q^{36} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{37} + ( 4 \zeta_{24}^{3} - 4 \zeta_{24}^{7} ) q^{38} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{39} + ( 8 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{40} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{41} + ( -1 - 4 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{42} + ( 6 + 2 \zeta_{24}^{3} + 6 \zeta_{24}^{6} ) q^{43} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{44} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{45} + ( -6 + 4 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{46} + ( -3 \zeta_{24} + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{47} + ( 4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{48} + ( -2 \zeta_{24} - 4 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( -8 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{50} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{51} + ( -2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{52} + ( -4 - 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{53} + ( -\zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{54} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{55} + ( 4 + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{56} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{57} + ( 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{58} + ( -5 - 5 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 5 \zeta_{24}^{7} ) q^{59} + ( -2 - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{60} + ( 4 \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{61} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{62} + ( -1 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( -5 \zeta_{24} - 6 \zeta_{24}^{4} - 5 \zeta_{24}^{7} ) q^{65} + ( 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{66} + ( -8 + 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{67} + ( -4 \zeta_{24} - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{68} + ( -3 + 4 \zeta_{24}^{3} - 3 \zeta_{24}^{6} ) q^{69} + ( -8 - 5 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 3 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{70} + ( -7 \zeta_{24} + 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{72} -4 \zeta_{24}^{2} q^{73} -4 \zeta_{24}^{5} q^{74} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{75} + ( 4 - 4 \zeta_{24}^{6} ) q^{76} + ( -2 - \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{77} + ( -2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{78} + ( 3 \zeta_{24} - 7 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{79} + ( 8 + 8 \zeta_{24}^{2} + 4 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{80} + ( 1 - \zeta_{24}^{4} ) q^{81} + ( -2 \zeta_{24} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{82} + ( -1 + 11 \zeta_{24} - 11 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( -2 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{84} + ( 10 + 12 \zeta_{24}^{3} + 10 \zeta_{24}^{6} ) q^{85} + ( 12 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{86} + ( 3 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{87} + ( 2 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} ) q^{88} + ( 11 \zeta_{24} - 11 \zeta_{24}^{7} ) q^{89} + ( -1 - 4 \zeta_{24}^{3} - \zeta_{24}^{6} ) q^{90} + ( -3 + \zeta_{24}^{2} - 4 \zeta_{24}^{3} - \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{91} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 8 \zeta_{24}^{6} ) q^{92} + ( \zeta_{24} - \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{93} + ( -6 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{94} + ( -8 - 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{95} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{96} + ( -3 - 8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{97} + ( -4 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{98} + ( -1 - \zeta_{24} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + 8q^{5} + 8q^{6} + O(q^{10}) \) \( 8q + 8q^{4} + 8q^{5} + 8q^{6} - 4q^{10} + 4q^{11} - 8q^{13} - 8q^{15} - 16q^{16} + 16q^{17} + 8q^{19} + 32q^{20} - 12q^{21} + 8q^{22} + 8q^{24} + 8q^{26} + 32q^{29} + 4q^{31} + 4q^{33} - 16q^{35} - 8q^{37} + 8q^{40} - 4q^{42} + 48q^{43} - 8q^{44} + 8q^{45} - 24q^{46} + 8q^{47} - 20q^{49} - 64q^{50} + 8q^{51} - 8q^{52} - 16q^{53} - 4q^{54} + 48q^{56} - 12q^{58} - 20q^{59} - 8q^{60} - 4q^{61} - 8q^{63} - 64q^{64} - 24q^{65} - 32q^{67} - 32q^{68} - 24q^{69} - 44q^{70} - 16q^{75} + 32q^{76} - 8q^{77} - 16q^{78} - 28q^{79} + 32q^{80} + 4q^{81} - 8q^{82} - 8q^{83} + 80q^{85} + 8q^{86} + 8q^{88} - 8q^{90} - 28q^{91} - 4q^{93} - 32q^{95} - 16q^{96} - 24q^{97} - 8q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(1\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−1.22474 0.707107i −0.965926 0.258819i 1.00000 + 1.73205i 3.69798 0.990870i 1.00000 + 1.00000i 0.358719 + 2.62132i 2.82843i 0.866025 + 0.500000i −5.22973 1.40130i
37.2 1.22474 + 0.707107i 0.965926 + 0.258819i 1.00000 + 1.73205i 1.76612 0.473232i 1.00000 + 1.00000i −2.09077 1.62132i 2.82843i 0.866025 + 0.500000i 2.49768 + 0.669251i
109.1 −1.22474 + 0.707107i −0.965926 + 0.258819i 1.00000 1.73205i 3.69798 + 0.990870i 1.00000 1.00000i 0.358719 2.62132i 2.82843i 0.866025 0.500000i −5.22973 + 1.40130i
109.2 1.22474 0.707107i 0.965926 0.258819i 1.00000 1.73205i 1.76612 + 0.473232i 1.00000 1.00000i −2.09077 + 1.62132i 2.82843i 0.866025 0.500000i 2.49768 0.669251i
205.1 −1.22474 0.707107i −0.258819 + 0.965926i 1.00000 + 1.73205i −0.473232 1.76612i 1.00000 1.00000i 2.09077 + 1.62132i 2.82843i −0.866025 0.500000i −0.669251 + 2.49768i
205.2 1.22474 + 0.707107i 0.258819 0.965926i 1.00000 + 1.73205i −0.990870 3.69798i 1.00000 1.00000i −0.358719 2.62132i 2.82843i −0.866025 0.500000i 1.40130 5.22973i
277.1 −1.22474 + 0.707107i −0.258819 0.965926i 1.00000 1.73205i −0.473232 + 1.76612i 1.00000 + 1.00000i 2.09077 1.62132i 2.82843i −0.866025 + 0.500000i −0.669251 2.49768i
277.2 1.22474 0.707107i 0.258819 + 0.965926i 1.00000 1.73205i −0.990870 + 3.69798i 1.00000 + 1.00000i −0.358719 + 2.62132i 2.82843i −0.866025 + 0.500000i 1.40130 + 5.22973i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes
16.e Even 1 yes
112.w Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{5}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).