Properties

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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{7} -\beta_{2} q^{9} + ( -1 + \beta_{1} - 2 \beta_{3} ) q^{11} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{13} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{15} + ( 4 - 4 \beta_{2} ) q^{17} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{19} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{21} + 4 \beta_{2} q^{23} + ( -10 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{25} + q^{27} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( -1 + \beta_{2} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( 8 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{35} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{43} + ( 1 - \beta_{1} + 2 \beta_{3} ) q^{45} -6 \beta_{2} q^{47} + ( -1 - 3 \beta_{1} + 3 \beta_{3} ) q^{49} + 4 \beta_{2} q^{51} + ( -5 - \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -14 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{59} -10 \beta_{2} q^{61} + ( -2 \beta_{2} - \beta_{3} ) q^{63} + ( -2 + 4 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 4 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{67} -4 q^{69} -2 q^{71} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{73} + ( 1 - 2 \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{75} + ( 3 + 2 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{77} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + ( 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 1 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{91} -\beta_{2} q^{93} + ( -12 - 2 \beta_{1} + 14 \beta_{2} + 4 \beta_{3} ) q^{95} + ( 12 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + q^{5} + 6q^{7} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + q^{5} + 6q^{7} - 2q^{9} - q^{11} + 10q^{13} - 2q^{15} + 8q^{17} - 5q^{19} - 3q^{21} + 8q^{23} - 19q^{25} + 4q^{27} + 6q^{29} - 2q^{31} - q^{33} + 30q^{35} - 3q^{37} - 5q^{39} + 12q^{41} + 14q^{43} + q^{45} - 12q^{47} - 10q^{49} + 8q^{51} - 11q^{53} - 58q^{55} + 10q^{57} + 5q^{59} - 20q^{61} - 3q^{63} - 26q^{65} + 7q^{67} - 16q^{69} - 8q^{71} + q^{73} - 19q^{75} + 27q^{77} + 8q^{79} - 2q^{81} - 14q^{83} + 8q^{85} - 3q^{87} + 6q^{89} + 15q^{91} - 2q^{93} - 26q^{95} + 50q^{97} + 2q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 5 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu + 5 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} + 4 \beta_{1} + 5\)

Character Values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0 −0.500000 + 0.866025i 0 −1.63746 2.83616i 0 1.50000 + 2.17945i 0 −0.500000 0.866025i 0
193.2 0 −0.500000 + 0.866025i 0 2.13746 + 3.70219i 0 1.50000 2.17945i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 −1.63746 + 2.83616i 0 1.50000 2.17945i 0 −0.500000 + 0.866025i 0
289.2 0 −0.500000 0.866025i 0 2.13746 3.70219i 0 1.50000 + 2.17945i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
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Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{4} - T_{5}^{3} + 15 T_{5}^{2} + 14 T_{5} + 196 \)
\( T_{11}^{4} + T_{11}^{3} + 15 T_{11}^{2} - 14 T_{11} + 196 \)