Embedded newform 336.2.bq.b.109.14

• Label: 336.2.bq.b.109.14
• Level: $336$
• Weight: $2$
• Character: 336.109
• 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			109.14
Character	\(\chi\)	\(=\)	336.109
Dual form			336.2.bq.b.37.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.244495 + 1.39292i) q^{2} +(0.965926 - 0.258819i) q^{3} +(-1.88044 - 0.681122i) q^{4} +(-3.28948 - 0.881413i) q^{5} +(0.124350 + 1.40874i) q^{6} +(2.12312 - 1.57873i) q^{7} +(1.40851 - 2.45278i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.244495	+	1.39292i	−0.172884	+	0.984942i
							\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
\(5\)	−3.28948	−	0.881413i	−1.47110	−	0.394180i								−0.567792	−	0.823172i	\(-0.692202\pi\)
							−0.903308	+	0.428992i	\(0.858869\pi\)
\(7\)	2.12312	−	1.57873i	0.802462	−	0.596703i
							\(11\)	−1.25556	−	4.68583i	−0.378567	−	1.41283i	−0.848062	−	0.529896i	\(-0.822231\pi\)
0.469495	−	0.882935i	\(-0.344436\pi\)
\(13\)	1.85406	+	1.85406i	0.514223	+	0.514223i	0.915818	−	0.401595i	\(-0.131544\pi\)
							−0.401595	+	0.915818i	\(0.631544\pi\)
\(17\)	1.09108	−	1.88981i	0.264627	−	0.458347i	−0.702839	−	0.711349i	\(-0.748085\pi\)
							0.967466	+	0.253002i	\(0.0814179\pi\)
\(19\)	0.974725	−	3.63772i	0.223617	−	0.834551i	−0.759336	−	0.650698i	\(-0.774476\pi\)
							0.982954	−	0.183853i	\(-0.0588570\pi\)
\(23\)	1.98367	−	1.14527i	0.413624	−	0.238806i	−0.278722	−	0.960372i	\(-0.589911\pi\)
							0.692346	+	0.721566i	\(0.256577\pi\)
\(29\)	−5.15275	−	5.15275i	−0.956841	−	0.956841i	0.0422650	−	0.999106i	\(-0.486543\pi\)
							−0.999106	+	0.0422650i	\(0.986543\pi\)
\(31\)	−4.18332	+	7.24573i	−0.751347	+	1.30137i	0.195823	+	0.980639i	\(0.437262\pi\)
							−0.947170	+	0.320732i	\(0.896071\pi\)
\(37\)	−5.63174	−	1.50902i	−0.925852	−	0.248081i	−0.235767	−	0.971810i	\(-0.575760\pi\)
							−0.690085	+	0.723728i	\(0.742427\pi\)
\(41\)		−	5.68268i		−	0.887485i	−0.896154	−	0.443743i	\(-0.853650\pi\)
							0.896154	−	0.443743i	\(-0.146350\pi\)
\(43\)	6.30422	−	6.30422i	0.961385	−	0.961385i	−0.0378969	−	0.999282i	\(-0.512066\pi\)
							0.999282	+	0.0378969i	\(0.0120658\pi\)
\(47\)	4.68925	+	8.12201i	0.683997	+	1.18472i	0.973751	+	0.227616i	\(0.0730932\pi\)
							−0.289754	+	0.957101i	\(0.593573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.109.14	yes	120
7.2	even	3	inner	336.2.bq.b.205.27	yes	120
16.5	even	4	inner	336.2.bq.b.277.27	yes	120
112.37	even	12	inner	336.2.bq.b.37.14	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.14	✓	120	112.37	even	12	inner
336.2.bq.b.109.14	yes	120	1.1	even	1	trivial
336.2.bq.b.205.27	yes	120	7.2	even	3	inner
336.2.bq.b.277.27	yes	120	16.5	even	4	inner