Properties

Label 336.2.q.c
Level 336
Weight 2
Character orbit 336.q
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.q (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} -3 q^{13} -2 q^{15} + ( -8 + 8 \zeta_{6} ) q^{17} -\zeta_{6} q^{19} + ( -1 - 2 \zeta_{6} ) q^{21} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + q^{27} + 4 q^{29} + ( 3 - 3 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} + ( -6 + 2 \zeta_{6} ) q^{35} + \zeta_{6} q^{37} + ( 3 - 3 \zeta_{6} ) q^{39} + 6 q^{41} -11 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} -8 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} + 4 q^{55} + q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} + 6 \zeta_{6} q^{61} + ( 3 - \zeta_{6} ) q^{63} -6 \zeta_{6} q^{65} + ( 13 - 13 \zeta_{6} ) q^{67} -8 q^{69} + 10 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 2 + 4 \zeta_{6} ) q^{77} -3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -2 q^{83} -16 q^{85} + ( -4 + 4 \zeta_{6} ) q^{87} + ( 6 - 9 \zeta_{6} ) q^{91} + 3 \zeta_{6} q^{93} + ( 2 - 2 \zeta_{6} ) q^{95} + 10 q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{5} - q^{7} - q^{9} + 2q^{11} - 6q^{13} - 4q^{15} - 8q^{17} - q^{19} - 4q^{21} + 8q^{23} + q^{25} + 2q^{27} + 8q^{29} + 3q^{31} + 2q^{33} - 10q^{35} + q^{37} + 3q^{39} + 12q^{41} - 22q^{43} + 2q^{45} + 6q^{47} - 13q^{49} - 8q^{51} + 12q^{53} + 8q^{55} + 2q^{57} + 4q^{59} + 6q^{61} + 5q^{63} - 6q^{65} + 13q^{67} - 16q^{69} + 20q^{71} + 11q^{73} + q^{75} + 8q^{77} - 3q^{79} - q^{81} - 4q^{83} - 32q^{85} - 4q^{87} + 3q^{91} + 3q^{93} + 2q^{95} + 20q^{97} - 4q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 1.00000 + 1.73205i 0 −0.500000 + 2.59808i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 1.00000 1.73205i 0 −0.500000 2.59808i 0 −0.500000 + 0.866025i 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
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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)