Properties

Label 336.2.k.a
Level 336
Weight 2
Character orbit 336.k
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.k (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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -1 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -5 + 4 \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} -12 q^{39} + 8 q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} + 6 q^{57} + ( -4 + 8 \zeta_{6} ) q^{61} + ( 3 + 6 \zeta_{6} ) q^{63} + 16 q^{67} + ( 8 - 16 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} + 4 q^{79} + 9 q^{81} + ( -20 + 16 \zeta_{6} ) q^{91} -18 q^{93} + ( -8 + 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} - 24q^{39} + 16q^{43} + 2q^{49} + 12q^{57} + 12q^{63} + 32q^{67} + 8q^{79} + 18q^{81} - 24q^{91} - 36q^{93} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 0 0 −2.00000 1.73205i 0 −3.00000 0
209.2 0 1.73205i 0 0 0 −2.00000 + 1.73205i 0 −3.00000 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
7.b Odd 1 yes
21.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).