Properties

Label 336.2.b.c
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: \(\Q(\sqrt{-6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta q^{5} + ( -1 + \beta ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta q^{5} + ( -1 + \beta ) q^{7} + q^{9} -\beta q^{11} + 2 \beta q^{13} + \beta q^{15} + \beta q^{17} -2 q^{19} + ( -1 + \beta ) q^{21} -3 \beta q^{23} - q^{25} + q^{27} + 6 q^{29} + 8 q^{31} -\beta q^{33} + ( -6 - \beta ) q^{35} + 4 q^{37} + 2 \beta q^{39} -3 \beta q^{41} + 2 \beta q^{43} + \beta q^{45} -12 q^{47} + ( -5 - 2 \beta ) q^{49} + \beta q^{51} -6 q^{53} + 6 q^{55} -2 q^{57} + 12 q^{59} + ( -1 + \beta ) q^{63} -12 q^{65} -3 \beta q^{69} -5 \beta q^{71} -6 \beta q^{73} - q^{75} + ( 6 + \beta ) q^{77} + 4 \beta q^{79} + q^{81} -6 q^{85} + 6 q^{87} -5 \beta q^{89} + ( -12 - 2 \beta ) q^{91} + 8 q^{93} -2 \beta q^{95} + 2 \beta q^{97} -\beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} - 4q^{19} - 2q^{21} - 2q^{25} + 2q^{27} + 12q^{29} + 16q^{31} - 12q^{35} + 8q^{37} - 24q^{47} - 10q^{49} - 12q^{53} + 12q^{55} - 4q^{57} + 24q^{59} - 2q^{63} - 24q^{65} - 2q^{75} + 12q^{77} + 2q^{81} - 12q^{85} + 12q^{87} - 24q^{91} + 16q^{93} + O(q^{100}) \)

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
2.44949i
2.44949i
0 1.00000 0 2.44949i 0 −1.00000 2.44949i 0 1.00000 0
223.2 0 1.00000 0 2.44949i 0 −1.00000 + 2.44949i 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 \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{2} + 6 \)
\( T_{19} + 2 \)