Embedded newform 336.2.bq.b.109.28

• Label: 336.2.bq.b.109.28
• 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			109.28
Character	\(\chi\)	\(=\)	336.109
Dual form			336.2.bq.b.37.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36230 + 0.379668i) q^{2} +(0.965926 - 0.258819i) q^{3} +(1.71170 + 1.03444i) q^{4} +(-0.180217 - 0.0482891i) q^{5} +(1.41414 + 0.0141427i) q^{6} +(-0.297360 - 2.62899i) q^{7} +(1.93911 + 2.05909i) q^{8} +(0.866025 - 0.500000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{4} - 8 q^{5} - 8 q^{6} + 24 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36230	+	0.379668i	0.963289	+	0.268466i
							\(3\)	0.965926	−	0.258819i	0.557678	−	0.149429i
\(5\)	−0.180217	−	0.0482891i	−0.0805956	−	0.0215955i								0.218296	−	0.975883i	\(-0.429950\pi\)
							−0.298892	+	0.954287i	\(0.596617\pi\)
\(7\)	−0.297360	−	2.62899i	−0.112391	−	0.993664i
							\(11\)	−0.954226	−	3.56122i	−0.287710	−	1.07375i	−0.946836	−	0.321716i	\(-0.895740\pi\)
0.659126	−	0.752032i	\(-0.270926\pi\)
\(13\)	1.93834	+	1.93834i	0.537598	+	0.537598i	0.922823	−	0.385225i	\(-0.125876\pi\)
							−0.385225	+	0.922823i	\(0.625876\pi\)
\(17\)	−2.59697	+	4.49809i	−0.629859	+	1.09095i	0.357721	+	0.933829i	\(0.383554\pi\)
							−0.987580	+	0.157119i	\(0.949779\pi\)
\(19\)	−2.15238	+	8.03278i	−0.493789	+	1.84285i	0.0429213	+	0.999078i	\(0.486334\pi\)
							−0.536710	+	0.843767i	\(0.680333\pi\)
\(23\)	−1.72648	+	0.996782i	−0.359995	+	0.207843i	−0.669079	−	0.743192i	\(-0.733311\pi\)
							0.309083	+	0.951035i	\(0.399978\pi\)
\(29\)	−5.48237	−	5.48237i	−1.01805	−	1.01805i	−0.999834	−	0.0182163i	\(-0.994201\pi\)
							−0.0182163	−	0.999834i	\(-0.505799\pi\)
\(31\)	2.96871	−	5.14195i	0.533196	−	0.923522i	−0.466053	−	0.884757i	\(-0.654324\pi\)
							0.999248	−	0.0387651i	\(-0.0123424\pi\)
\(37\)	2.21208	+	0.592726i	0.363664	+	0.0974435i	0.436024	−	0.899935i	\(-0.356386\pi\)
							−0.0723598	+	0.997379i	\(0.523053\pi\)
\(41\)			7.26176i			1.13410i	0.823685	+	0.567048i	\(0.191915\pi\)
							−0.823685	+	0.567048i	\(0.808085\pi\)
\(43\)	4.07117	−	4.07117i	0.620847	−	0.620847i	−0.324901	−	0.945748i	\(-0.605331\pi\)
							0.945748	+	0.324901i	\(0.105331\pi\)
\(47\)	−6.35419	−	11.0058i	−0.926853	−	1.60536i	−0.788553	−	0.614967i	\(-0.789169\pi\)
							−0.138300	−	0.990390i	\(-0.544164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.2.bq.b.109.28	yes	120
7.2	even	3	inner	336.2.bq.b.205.12	yes	120
16.5	even	4	inner	336.2.bq.b.277.12	yes	120
112.37	even	12	inner	336.2.bq.b.37.28	✓	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bq.b.37.28	✓	120	112.37	even	12	inner
336.2.bq.b.109.28	yes	120	1.1	even	1	trivial
336.2.bq.b.205.12	yes	120	7.2	even	3	inner
336.2.bq.b.277.12	yes	120	16.5	even	4	inner