Properties

Label 336.2.b.a
Level 336
Weight 2
Character orbit 336.b
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} + q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 4 - 8 \zeta_{6} ) q^{17} -4 q^{19} + ( 1 + 2 \zeta_{6} ) q^{21} + ( 2 - 4 \zeta_{6} ) q^{23} + 5 q^{25} - q^{27} -6 q^{29} + 4 q^{31} + ( -2 + 4 \zeta_{6} ) q^{33} -2 q^{37} + ( 4 - 8 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -4 + 8 \zeta_{6} ) q^{51} -6 q^{53} + 4 q^{57} + 12 q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + ( -1 - 2 \zeta_{6} ) q^{63} + ( -2 + 4 \zeta_{6} ) q^{67} + ( -2 + 4 \zeta_{6} ) q^{69} + ( -6 + 12 \zeta_{6} ) q^{71} + ( 8 - 16 \zeta_{6} ) q^{73} -5 q^{75} + ( -10 + 8 \zeta_{6} ) q^{77} + ( -6 + 12 \zeta_{6} ) q^{79} + q^{81} + 12 q^{83} + 6 q^{87} + ( -4 + 8 \zeta_{6} ) q^{89} -4 q^{93} + ( 2 - 4 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{7} + 2q^{9} - 8q^{19} + 4q^{21} + 10q^{25} - 2q^{27} - 12q^{29} + 8q^{31} - 4q^{37} + 2q^{49} - 12q^{53} + 8q^{57} + 24q^{59} - 4q^{63} - 10q^{75} - 12q^{77} + 2q^{81} + 24q^{83} + 12q^{87} - 8q^{93} + O(q^{100}) \)

Character Values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.00000 0 0 0 −2.00000 1.73205i 0 1.00000 0
223.2 0 −1.00000 0 0 0 −2.00000 + 1.73205i 0 1.00000 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
28.d Even 1 yes

Hecke kernels

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

\( T_{5} \)
\( T_{19} + 4 \)