Properties

Label 336.2.k.b
Level 336
Weight 2
Character orbit 336.k
Analytic conductor 2.683
Analytic rank 0
Dimension 4
CM No
Inner twists 4

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} ) q^{7} + \beta_{2} q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( 3 + \beta_{2} ) q^{15} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + \beta_{1} - \beta_{2} ) q^{21} + 2 \beta_{2} q^{23} + q^{25} + 3 \beta_{3} q^{27} + 2 \beta_{2} q^{29} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} -2 q^{37} + ( 3 - \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} -4 q^{43} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -6 - 2 \beta_{2} ) q^{51} -2 \beta_{2} q^{53} + ( 3 - \beta_{2} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{59} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{63} -2 \beta_{2} q^{65} -8 q^{67} + 6 \beta_{3} q^{69} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{73} + \beta_{1} q^{75} + 10 q^{79} -9 q^{81} + ( \beta_{1} - \beta_{3} ) q^{83} -12 q^{85} + 6 \beta_{3} q^{87} + ( -6 - \beta_{1} - \beta_{3} ) q^{91} -2 \beta_{2} q^{95} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} + 12q^{15} + 12q^{21} + 4q^{25} - 8q^{37} + 12q^{39} - 16q^{43} - 20q^{49} - 24q^{51} + 12q^{57} - 32q^{67} + 40q^{79} - 36q^{81} - 48q^{85} - 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

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
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
0 −1.22474 1.22474i 0 −2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.2 0 −1.22474 + 1.22474i 0 −2.44949 0 1.00000 2.44949i 0 3.00000i 0
209.3 0 1.22474 1.22474i 0 2.44949 0 1.00000 + 2.44949i 0 3.00000i 0
209.4 0 1.22474 + 1.22474i 0 2.44949 0 1.00000 2.44949i 0 3.00000i 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 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 \( T_{5}^{2} - 6 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).