Properties

Label 336.2.bc.a
Level 336
Weight 2
Character orbit 336.bc
Analytic conductor 2.683
Analytic rank 0
Dimension 2
CM disc. -3
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + ( 2 + \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 6 \zeta_{6} ) q^{13} + ( 10 - 5 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} + ( 5 - 5 \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 + \zeta_{6} ) q^{31} + 11 \zeta_{6} q^{37} + ( -9 + 9 \zeta_{6} ) q^{39} -13 q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} -15 q^{57} + ( -8 + 4 \zeta_{6} ) q^{61} + ( -3 + 9 \zeta_{6} ) q^{63} + ( 5 - 5 \zeta_{6} ) q^{67} + ( -1 - \zeta_{6} ) q^{73} + ( -10 + 5 \zeta_{6} ) q^{75} + 17 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 15 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 5q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 5q^{7} + 3q^{9} + 15q^{19} - 6q^{21} + 5q^{25} + 3q^{31} + 11q^{37} - 9q^{39} - 26q^{43} + 11q^{49} - 30q^{57} - 12q^{61} + 3q^{63} + 5q^{67} - 3q^{73} - 15q^{75} + 17q^{79} - 9q^{81} + 9q^{91} - 3q^{93} + O(q^{100}) \)

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 −1.50000 0.866025i 0 0 0 2.50000 + 0.866025i 0 1.50000 + 2.59808i 0
257.1 0 −1.50000 + 0.866025i 0 0 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 CM by \(\Q(\sqrt{-3}) \) 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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5} \)
\( T_{13}^{2} + 27 \)