Embedded newform 349.3.f.a.24.13

• Label: 349.3.f.a.24.13
• Level: $349$
• Weight: $3$
• Character: 349.24
• Analytic conductor: $9.510$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 349 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	349.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.50956122617\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(57\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			24.13
Character	\(\chi\)	\(=\)	349.24
Dual form			349.3.f.a.160.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38235 - 0.638349i) q^{2} +(-1.90304 - 1.09872i) q^{3} +(1.80400 + 1.04154i) q^{4} +(4.02016 + 2.32104i) q^{5} +(3.83233 + 3.83233i) q^{6} +(-2.80885 + 10.4828i) q^{7} +(3.34311 + 3.34311i) q^{8} +(-2.08564 - 3.61243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} + 6 q^{4} - 38 q^{6} + 22 q^{7} - 12 q^{8} + 322 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38235	−	0.638349i	−1.19118	−	0.319174i	−0.391825	−	0.920040i	\(-0.628156\pi\)
							−0.799350	+	0.600865i	\(0.794823\pi\)
\(3\)	−1.90304	−	1.09872i	−0.634345	−	0.366239i	0.148088	−	0.988974i	\(-0.452688\pi\)
							−0.782433	+	0.622735i	\(0.786021\pi\)
\(5\)	4.02016	+	2.32104i	0.804032	+	0.464208i	0.844879	−	0.534958i	\(-0.179672\pi\)
							−0.0408474	+	0.999165i	\(0.513006\pi\)
\(7\)	−2.80885	+	10.4828i	−0.401264	+	1.49754i	0.409580	+	0.912274i	\(0.365675\pi\)
							−0.810844	+	0.585263i	\(0.800991\pi\)
\(11\)	−0.497894	+	0.497894i	−0.0452631	+	0.0452631i	−0.729376	−	0.684113i	\(-0.760190\pi\)
							0.684113	+	0.729376i	\(0.260190\pi\)
\(13\)	−0.747736	+	2.79059i	−0.0575181	+	0.214661i	−0.988703	−	0.149886i	\(-0.952109\pi\)
							0.931185	+	0.364547i	\(0.118776\pi\)
\(17\)			17.9510i			1.05594i	0.849263	+	0.527970i	\(0.177047\pi\)
							−0.849263	+	0.527970i	\(0.822953\pi\)
\(19\)	13.6067	−	23.5675i	0.716143	−	1.24040i	−0.246374	−	0.969175i	\(-0.579239\pi\)
							0.962517	−	0.271222i	\(-0.0874276\pi\)
\(23\)	−14.6772	−	25.4217i	−0.638141	−	1.10529i	−0.985840	−	0.167686i	\(-0.946370\pi\)
							0.347700	−	0.937606i	\(-0.386963\pi\)
\(29\)	14.3742	+	8.29895i	0.495662	+	0.286171i	0.726921	−	0.686722i	\(-0.240951\pi\)
							−0.231258	+	0.972892i	\(0.574284\pi\)
\(31\)	−42.9772			−1.38636			−0.693181	−	0.720763i	\(-0.743791\pi\)
							−0.693181	+	0.720763i	\(0.743791\pi\)
\(37\)			37.8136i			1.02199i	0.859584	+	0.510994i	\(0.170723\pi\)
							−0.859584	+	0.510994i	\(0.829277\pi\)
\(41\)	−26.0183			−0.634592			−0.317296	−	0.948327i	\(-0.602775\pi\)
							−0.317296	+	0.948327i	\(0.602775\pi\)
\(43\)	4.38420	+	16.3620i	0.101958	+	0.380513i	0.997982	−	0.0634924i	\(-0.0202239\pi\)
							−0.896024	+	0.444005i	\(0.853557\pi\)
\(47\)	−48.9454	−	48.9454i	−1.04139	−	1.04139i	−0.999106	−	0.0422866i	\(-0.986536\pi\)
							−0.0422866	−	0.999106i	\(-0.513464\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	349.3.f.a.24.13	✓	228
349.160	odd	12	inner	349.3.f.a.160.13	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
349.3.f.a.24.13	✓	228	1.1	even	1	trivial
349.3.f.a.160.13	yes	228	349.160	odd	12	inner