Properties

Label 336.2.k.c
Level 336
Weight 2
Character orbit 336.k
Analytic conductor 2.683
Analytic rank 0
Dimension 8
CM No
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: \(8\)
Coefficient field: 8.0.342102016.5
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( -\beta_{5} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( -\beta_{5} - \beta_{7} ) q^{9} + ( -\beta_{3} + \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( 3 \beta_{1} - 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{21} + \beta_{3} q^{23} + ( 1 + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{27} + ( -2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{35} -2 q^{37} + ( -1 + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{39} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{41} + 4 q^{43} + ( -2 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{45} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} ) q^{47} + ( 3 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{49} + ( 2 - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{51} + ( 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{53} + ( -4 \beta_{1} + 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{55} + ( -5 + 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{59} + ( -3 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{61} + ( -4 - 2 \beta_{1} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{63} + ( -\beta_{6} + \beta_{7} ) q^{65} + ( 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{69} + ( 5 \beta_{3} - \beta_{6} + \beta_{7} ) q^{71} + ( -4 \beta_{1} + 4 \beta_{4} ) q^{73} + ( \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{75} + ( -3 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{77} + ( -6 + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{79} + ( -1 - 4 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{81} + ( \beta_{1} + \beta_{4} ) q^{83} + 4 q^{85} + ( -2 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{87} + ( -5 \beta_{1} + \beta_{2} - 5 \beta_{4} ) q^{89} + ( 2 + \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 8 + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{93} + ( -2 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{95} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + ( -8 - 5 \beta_{3} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{7} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{7} + 4q^{9} - 4q^{15} - 8q^{21} - 16q^{37} - 4q^{39} + 32q^{43} + 16q^{49} + 24q^{51} - 28q^{57} - 32q^{63} - 16q^{67} - 56q^{79} + 32q^{85} + 24q^{91} + 56q^{93} - 64q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8 \)\()/16\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 2 \nu^{3} \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + \nu^{5} + 2 \nu^{3} \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + \nu^{5} - 3 \nu^{4} - 6 \nu^{3} - 10 \nu^{2} + 8 \nu - 8 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} - 3 \nu^{4} + 6 \nu^{2} - 8 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} + 10 \nu^{3} - 12 \nu^{2} + 24 \nu \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 3 \nu^{5} + 2 \nu^{4} - 10 \nu^{3} - 12 \nu^{2} - 24 \nu \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{1} - 4\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 3 \beta_{4} + 5 \beta_{3} + 9 \beta_{2} + 3 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-6 \beta_{7} - 6 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - 3 \beta_{1} - 4\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - 3 \beta_{4} - 21 \beta_{3} + 7 \beta_{2} - 3 \beta_{1}\)\()/4\)

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
−0.599676 + 1.28078i
−0.599676 1.28078i
−1.17915 + 0.780776i
−1.17915 0.780776i
1.17915 0.780776i
1.17915 + 0.780776i
0.599676 1.28078i
0.599676 + 1.28078i
0 −1.66757 0.468213i 0 −0.936426 0 −1.56155 2.13578i 0 2.56155 + 1.56155i 0
209.2 0 −1.66757 + 0.468213i 0 −0.936426 0 −1.56155 + 2.13578i 0 2.56155 1.56155i 0
209.3 0 −0.848071 1.51022i 0 3.02045 0 2.56155 0.662153i 0 −1.56155 + 2.56155i 0
209.4 0 −0.848071 + 1.51022i 0 3.02045 0 2.56155 + 0.662153i 0 −1.56155 2.56155i 0
209.5 0 0.848071 1.51022i 0 −3.02045 0 2.56155 0.662153i 0 −1.56155 2.56155i 0
209.6 0 0.848071 + 1.51022i 0 −3.02045 0 2.56155 + 0.662153i 0 −1.56155 + 2.56155i 0
209.7 0 1.66757 0.468213i 0 0.936426 0 −1.56155 2.13578i 0 2.56155 1.56155i 0
209.8 0 1.66757 + 0.468213i 0 0.936426 0 −1.56155 + 2.13578i 0 2.56155 + 1.56155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
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 \( T_{5}^{4} - 10 T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).