Properties

Label 336.2.bc.b
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
Analytic rank 0
Dimension 2
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.bc (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -3 - 3 \zeta_{6} ) q^{11} + ( 6 - 3 \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( -2 + \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( 6 - 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 + \zeta_{6} ) q^{31} + ( 9 - 9 \zeta_{6} ) q^{33} + ( -6 + 9 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + 6 q^{41} -4 q^{43} + 9 \zeta_{6} q^{45} + 3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -3 - 3 \zeta_{6} ) q^{51} + ( -3 - 3 \zeta_{6} ) q^{53} + ( -9 + 18 \zeta_{6} ) q^{55} -3 \zeta_{6} q^{57} + ( -3 + 3 \zeta_{6} ) q^{59} + ( -14 + 7 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + 9 \zeta_{6} q^{69} + ( 6 - 12 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( -4 - 4 \zeta_{6} ) q^{75} + ( -3 + 15 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 9 q^{81} + 12 q^{83} + 9 q^{85} + 9 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{93} + ( 3 + 3 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( 9 + 9 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} - 4q^{7} - 6q^{9} - 9q^{11} + 9q^{15} - 3q^{17} - 3q^{19} + 6q^{21} + 9q^{23} - 4q^{25} + 3q^{31} + 9q^{33} - 3q^{35} - 7q^{37} + 12q^{41} - 8q^{43} + 9q^{45} + 3q^{47} + 2q^{49} - 9q^{51} - 9q^{53} - 3q^{57} - 3q^{59} - 21q^{61} + 12q^{63} + 5q^{67} + 9q^{69} - 21q^{73} - 12q^{75} + 9q^{77} - q^{79} + 18q^{81} + 24q^{83} + 18q^{85} + 9q^{89} - 3q^{93} + 9q^{95} + 27q^{99} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −1.50000 2.59808i 0 −2.00000 1.73205i 0 −3.00000 0
257.1 0 1.73205i 0 −1.50000 + 2.59808i 0 −2.00000 + 1.73205i 0 −3.00000 0
\(n\): 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
21.g Even 1 yes

Hecke kernels

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

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{13} \)