Properties

Label 336.2.bc.e
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
Analytic rank 0
Dimension 4
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.bc (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 + \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{5} + ( 2 + \zeta_{12}^{2} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 q^{15} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -2 - 2 \zeta_{12}^{2} ) q^{19} + ( -\zeta_{12} - 4 \zeta_{12}^{3} ) q^{21} -6 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 3 \zeta_{12}^{3} q^{29} + ( 2 - \zeta_{12}^{2} ) q^{31} + ( -3 - 3 \zeta_{12}^{2} ) q^{33} + ( 4 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{35} + ( 2 - 2 \zeta_{12}^{2} ) q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{39} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + 8 q^{43} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{45} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{47} + ( 3 + 5 \zeta_{12}^{2} ) q^{49} + ( -6 + 6 \zeta_{12}^{2} ) q^{51} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{59} + ( -9 + 6 \zeta_{12}^{2} ) q^{63} -6 \zeta_{12} q^{65} + 2 \zeta_{12}^{2} q^{67} + ( -6 + 12 \zeta_{12}^{2} ) q^{69} + 12 \zeta_{12}^{3} q^{71} + ( 8 - 4 \zeta_{12}^{2} ) q^{73} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{75} + ( 9 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{77} + ( -1 + \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} -6 q^{85} + ( 6 - 3 \zeta_{12}^{2} ) q^{87} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( 8 - 10 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12} q^{93} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{95} + ( 3 - 6 \zeta_{12}^{2} ) q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 10q^{7} - 6q^{9} - 12q^{15} - 12q^{19} + 4q^{25} + 6q^{31} - 18q^{33} + 4q^{37} + 32q^{43} + 22q^{49} - 12q^{51} - 24q^{63} + 4q^{67} + 24q^{73} - 2q^{79} - 18q^{81} - 24q^{85} + 18q^{87} + 12q^{91} + O(q^{100}) \)

Character Values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 - \zeta_{12}\)

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
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 0.866025 + 1.50000i 0 2.50000 0.866025i 0 −1.50000 2.59808i 0
17.2 0 0.866025 1.50000i 0 −0.866025 1.50000i 0 2.50000 0.866025i 0 −1.50000 2.59808i 0
257.1 0 −0.866025 1.50000i 0 0.866025 1.50000i 0 2.50000 + 0.866025i 0 −1.50000 + 2.59808i 0
257.2 0 0.866025 + 1.50000i 0 −0.866025 + 1.50000i 0 2.50000 + 0.866025i 0 −1.50000 + 2.59808i 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
3.b Odd 1 yes
7.d Odd 1 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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{4} + 3 T_{5}^{2} + 9 \)
\( T_{13}^{2} + 12 \)