Properties

Label 336.2.bl.g
Level 336
Weight 2
Character orbit 336.bl
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.bl (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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} + ( 2 - \zeta_{6} ) q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 - \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{15} + ( 2 + 2 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} + ( -2 - \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{25} - q^{27} + 9 q^{29} + ( 5 - 5 \zeta_{6} ) q^{31} + ( -2 + \zeta_{6} ) q^{33} + ( -1 - 4 \zeta_{6} ) q^{35} -10 \zeta_{6} q^{37} + ( -6 + 12 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -1 - \zeta_{6} ) q^{45} + 12 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 4 - 2 \zeta_{6} ) q^{51} + ( 9 - 9 \zeta_{6} ) q^{53} -3 q^{55} -2 q^{57} + ( -9 + 9 \zeta_{6} ) q^{59} + ( -3 + 2 \zeta_{6} ) q^{63} + ( 8 + 8 \zeta_{6} ) q^{67} + ( -8 + 16 \zeta_{6} ) q^{71} + ( -4 - 4 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{75} + ( -4 + 5 \zeta_{6} ) q^{77} + ( -6 + 3 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -3 q^{83} + 6 q^{85} + ( 9 - 9 \zeta_{6} ) q^{87} + ( -4 + 2 \zeta_{6} ) q^{89} -5 \zeta_{6} q^{93} + ( -2 - 2 \zeta_{6} ) q^{95} + ( -11 + 22 \zeta_{6} ) q^{97} + ( -1 + 2 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 3q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 3q^{5} - q^{7} - q^{9} - 3q^{11} + 6q^{17} - 2q^{19} - 5q^{21} - 2q^{25} - 2q^{27} + 18q^{29} + 5q^{31} - 3q^{33} - 6q^{35} - 10q^{37} - 3q^{45} + 12q^{47} - 13q^{49} + 6q^{51} + 9q^{53} - 6q^{55} - 4q^{57} - 9q^{59} - 4q^{63} + 24q^{67} - 12q^{73} + 2q^{75} - 3q^{77} - 9q^{79} - q^{81} - 6q^{83} + 12q^{85} + 9q^{87} - 6q^{89} - 5q^{93} - 6q^{95} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 0.866025i 0 1.50000 0.866025i 0 −0.500000 2.59808i 0 −0.500000 0.866025i 0
271.1 0 0.500000 + 0.866025i 0 1.50000 + 0.866025i 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
28.f Even 1 yes

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 3 \)
\( T_{11}^{2} + 3 T_{11} + 3 \)