Properties

Label 336.2.bj.f
Level 336
Weight 2
Character orbit 336.bj
Analytic conductor 2.683
Analytic rank 0
Dimension 8
CM No
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 336.bj (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{5} + ( \beta_{3} + 3 \beta_{5} ) q^{7} + ( -2 - \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - 2 \beta_{2} + \beta_{4} + 2 \beta_{6} ) q^{5} + ( \beta_{3} + 3 \beta_{5} ) q^{7} + ( -2 - \beta_{2} - 2 \beta_{4} + \beta_{6} ) q^{9} + ( -2 \beta_{1} + \beta_{5} ) q^{11} -4 q^{13} + ( \beta_{1} - 6 \beta_{3} - 6 \beta_{5} - \beta_{7} ) q^{15} + ( -2 \beta_{2} + \beta_{4} ) q^{17} + 7 \beta_{3} q^{19} + ( 2 - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{21} + ( \beta_{3} + 2 \beta_{7} ) q^{23} -6 \beta_{4} q^{25} + ( -2 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{27} + ( 2 + 4 \beta_{6} ) q^{29} + 3 \beta_{5} q^{31} + ( 5 + \beta_{2} + 5 \beta_{4} - \beta_{6} ) q^{33} + ( 4 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} - 6 \beta_{7} ) q^{35} + ( 1 + \beta_{4} ) q^{37} -4 \beta_{1} q^{39} + ( -2 - 4 \beta_{6} ) q^{41} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{43} + ( 5 \beta_{2} - 8 \beta_{4} ) q^{45} + ( 3 \beta_{3} + 6 \beta_{7} ) q^{47} + ( 3 + 8 \beta_{4} ) q^{49} + ( -6 \beta_{3} - \beta_{7} ) q^{51} + ( 2 \beta_{2} - \beta_{4} ) q^{53} + ( 11 \beta_{3} + 11 \beta_{5} ) q^{55} + ( -7 - 7 \beta_{6} ) q^{57} + ( 2 \beta_{1} - \beta_{5} ) q^{59} + ( -3 - 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} - 9 \beta_{3} - 6 \beta_{5} - 3 \beta_{7} ) q^{63} + ( -4 + 8 \beta_{2} - 4 \beta_{4} - 8 \beta_{6} ) q^{65} + 9 \beta_{5} q^{67} + ( -5 + \beta_{6} ) q^{69} + ( 8 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} ) q^{71} + 7 \beta_{4} q^{73} + 6 \beta_{7} q^{75} + ( -2 + 6 \beta_{2} - 3 \beta_{4} - 4 \beta_{6} ) q^{77} + 9 \beta_{3} q^{79} + ( 5 \beta_{2} + \beta_{4} ) q^{81} + ( -8 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} + 8 \beta_{7} ) q^{83} + 11 q^{85} + ( 2 \beta_{1} - 12 \beta_{5} ) q^{87} + ( -1 + 2 \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{89} + ( -4 \beta_{3} - 12 \beta_{5} ) q^{91} + ( 3 - 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{6} ) q^{93} + ( -14 \beta_{1} + 7 \beta_{5} ) q^{95} + 8 q^{97} + ( 5 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} - 5 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 10q^{9} + O(q^{10}) \) \( 8q - 10q^{9} - 32q^{13} + 2q^{21} + 24q^{25} + 22q^{33} + 4q^{37} + 22q^{45} - 8q^{49} - 28q^{57} - 12q^{61} - 44q^{69} - 28q^{73} - 14q^{81} + 88q^{85} + 6q^{93} + 64q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 32 \nu^{4} + 16 \nu^{2} + 45 \)\()/144\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 32 \nu^{5} + 16 \nu^{3} + 45 \nu \)\()/432\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{6} - 16 \nu^{4} - 80 \nu^{2} - 225 \)\()/144\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/48\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 13 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 16 \nu^{5} + 80 \nu^{3} + 225 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{4} - \beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 3 \beta_{5} - 3 \beta_{3} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + 15 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 13\)
\(\nu^{7}\)\(=\)\(48 \beta_{5} - 13 \beta_{1}\)

Character Values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
−1.26217 + 1.18614i
−0.396143 + 1.68614i
0.396143 1.68614i
1.26217 1.18614i
−1.26217 1.18614i
−0.396143 1.68614i
0.396143 + 1.68614i
1.26217 + 1.18614i
0 −1.26217 + 1.18614i 0 2.87228 1.65831i 0 −1.73205 2.00000i 0 0.186141 2.99422i 0
95.2 0 −0.396143 + 1.68614i 0 −2.87228 + 1.65831i 0 1.73205 + 2.00000i 0 −2.68614 1.33591i 0
95.3 0 0.396143 1.68614i 0 −2.87228 + 1.65831i 0 −1.73205 2.00000i 0 −2.68614 1.33591i 0
95.4 0 1.26217 1.18614i 0 2.87228 1.65831i 0 1.73205 + 2.00000i 0 0.186141 2.99422i 0
191.1 0 −1.26217 1.18614i 0 2.87228 + 1.65831i 0 −1.73205 + 2.00000i 0 0.186141 + 2.99422i 0
191.2 0 −0.396143 1.68614i 0 −2.87228 1.65831i 0 1.73205 2.00000i 0 −2.68614 + 1.33591i 0
191.3 0 0.396143 + 1.68614i 0 −2.87228 1.65831i 0 −1.73205 + 2.00000i 0 −2.68614 + 1.33591i 0
191.4 0 1.26217 + 1.18614i 0 2.87228 + 1.65831i 0 1.73205 2.00000i 0 0.186141 + 2.99422i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
4.b Odd 1 yes
7.c Even 1 yes
12.b Even 1 yes
21.h Odd 1 yes
28.g Odd 1 yes
84.n Even 1 yes

Hecke kernels

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

\( T_{5}^{4} - 11 T_{5}^{2} + 121 \)
\( T_{13} + 4 \)
\( T_{19}^{4} - 49 T_{19}^{2} + 2401 \)