Embedded newform 336.2.bq.b.37.9

• Label: 336.2.bq.b.37.9
• Level: $336$
• Weight: $2$
• Character: 336.37
• Analytic conductor: $2.683$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	336.bq (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.68297350792\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.9
Character	\(\chi\)	\(=\)	336.37
Dual form			336.2.bq.b.109.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.975065 - 1.02433i) q^{2} +(0.965926 + 0.258819i) q^{3} +(-0.0984971 + 1.99757i) q^{4} +(3.52312 - 0.944018i) q^{5} +(-0.676725 - 1.24179i) q^{6} +(-0.745861 - 2.53844i) q^{7} +(2.14221 - 1.84687i) q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.975065	−	1.02433i	−0.689475	−	0.724310i
							\(3\)	0.965926	+	0.258819i	0.557678	+	0.149429i
\(5\)	3.52312	−	0.944018i	1.57559	−	0.422178i								0.638032	−	0.770010i	\(-0.279749\pi\)
							0.937557	+	0.347832i	\(0.113082\pi\)
\(7\)	−0.745861	−	2.53844i	−0.281909	−	0.959441i
							\(11\)	−1.38883	+	5.18320i	−0.418749	+	1.56279i	0.358457	+	0.933546i	\(0.383303\pi\)
−0.777206	+	0.629246i	\(0.783364\pi\)
\(13\)	2.26482	−	2.26482i	0.628149	−	0.628149i	−0.319453	−	0.947602i	\(-0.603499\pi\)
							0.947602	+	0.319453i	\(0.103499\pi\)
\(17\)	−1.80843	−	3.13228i	−0.438608	−	0.759690i	0.558975	−	0.829185i	\(-0.311195\pi\)
							−0.997582	+	0.0694941i	\(0.977861\pi\)
\(19\)	0.531626	+	1.98406i	0.121963	+	0.455174i	0.999713	−	0.0239455i	\(-0.00762282\pi\)
							−0.877750	+	0.479119i	\(0.840956\pi\)
\(23\)	4.55873	+	2.63199i	0.950561	+	0.548807i	0.893255	−	0.449550i	\(-0.148416\pi\)
							0.0573061	+	0.998357i	\(0.481749\pi\)
\(29\)	−4.19993	+	4.19993i	−0.779906	+	0.779906i	−0.979815	−	0.199908i	\(-0.935936\pi\)
							0.199908	+	0.979815i	\(0.435936\pi\)
\(31\)	−4.43270	−	7.67766i	−0.796136	−	1.37895i	−0.922115	−	0.386915i	\(-0.873541\pi\)
							0.125979	−	0.992033i	\(-0.459793\pi\)
\(37\)	4.83027	−	1.29427i	0.794092	−	0.212776i	0.161103	−	0.986938i	\(-0.448495\pi\)
							0.632988	+	0.774161i	\(0.281828\pi\)
\(41\)		−	6.38018i		−	0.996417i	−0.867057	−	0.498208i	\(-0.833991\pi\)
							0.867057	−	0.498208i	\(-0.166009\pi\)
\(43\)	−3.19092	−	3.19092i	−0.486610	−	0.486610i	0.420625	−	0.907235i	\(-0.361811\pi\)
							−0.907235	+	0.420625i	\(0.861811\pi\)
\(47\)	−1.43670	+	2.48845i	−0.209565	+	0.362977i	−0.951578	−	0.307409i	\(-0.900538\pi\)
							0.742013	+	0.670386i	\(0.233871\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.37.9	✓	120
7.4	even	3	inner	336.2.bq.b.277.28	yes	120
16.13	even	4	inner	336.2.bq.b.205.28	yes	120
112.109	even	12	inner	336.2.bq.b.109.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.9	✓	120	1.1	even	1	trivial
336.2.bq.b.109.9	yes	120	112.109	even	12	inner
336.2.bq.b.205.28	yes	120	16.13	even	4	inner
336.2.bq.b.277.28	yes	120	7.4	even	3	inner