Embedded newform 211.3.m.a.138.14

• Label: 211.3.m.a.138.14
• Level: $211$
• Weight: $3$
• Character: 211.138
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $408$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 211 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	211.m (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74933357800\)
Analytic rank:	\(0\)
Dimension:	\(408\)
Relative dimension:	\(34\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			138.14
Character	\(\chi\)	\(=\)	211.138
Dual form			211.3.m.a.26.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.736841 - 1.08075i) q^{2} +(2.42558 + 3.55767i) q^{3} +(0.836285 - 2.13082i) q^{4} +(-5.35663 + 2.57962i) q^{5} +(2.05767 - 5.24287i) q^{6} +(-6.25866 - 0.469022i) q^{7} +(-8.02003 + 1.83052i) q^{8} +(-3.48550 + 8.88091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 408 q - 4 q^{2} - 14 q^{3} - 90 q^{4} - 10 q^{5} + 29 q^{6} + 10 q^{7} + 56 q^{8} - 96 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/211\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.736841	−	1.08075i	−0.368420	−	0.540373i	0.596603	−	0.802536i	\(-0.296517\pi\)
							−0.965024	+	0.262163i	\(0.915564\pi\)
\(3\)	2.42558	+	3.55767i	0.808525	+	1.18589i	0.979665	+	0.200642i	\(0.0643027\pi\)
							−0.171140	+	0.985247i	\(0.554745\pi\)
\(5\)	−5.35663	+	2.57962i	−1.07133	+	0.515924i	−0.884534	−	0.466476i	\(-0.845523\pi\)
							−0.186792	+	0.982399i	\(0.559809\pi\)
\(7\)	−6.25866	−	0.469022i	−0.894094	−	0.0670031i	−0.380247	−	0.924885i	\(-0.624161\pi\)
							−0.513847	+	0.857882i	\(0.671780\pi\)
\(11\)	−11.1794	−	5.38374i	−1.01631	−	0.489431i	−0.149870	−	0.988706i	\(-0.547885\pi\)
							−0.866444	+	0.499275i	\(0.833600\pi\)
\(13\)	−4.85802	+	21.2844i	−0.373694	+	1.63726i	0.342614	+	0.939476i	\(0.388688\pi\)
							−0.716308	+	0.697784i	\(0.754169\pi\)
\(17\)	6.05965	−	19.6449i	0.356450	−	1.15558i	−0.582877	−	0.812561i	\(-0.698073\pi\)
							0.939327	−	0.343023i	\(-0.111451\pi\)
\(19\)	−2.89748	+	5.01858i	−0.152499	+	0.264136i	−0.932145	−	0.362084i	\(-0.882065\pi\)
							0.779647	+	0.626220i	\(0.215399\pi\)
\(23\)		−	2.71511i		−	0.118048i	−0.998257	−	0.0590242i	\(-0.981201\pi\)
							0.998257	−	0.0590242i	\(-0.0187989\pi\)
\(29\)	−13.7428	−	44.5531i	−0.473890	−	1.53631i	−0.805323	−	0.592836i	\(-0.798008\pi\)
							0.331433	−	0.943479i	\(-0.392468\pi\)
\(31\)	−17.7516	+	57.5492i	−0.572631	+	1.85642i	−0.0592881	+	0.998241i	\(0.518883\pi\)
							−0.513343	+	0.858184i	\(0.671593\pi\)
\(37\)	−2.10436	+	28.0807i	−0.0568746	+	0.758939i	0.893744	+	0.448577i	\(0.148069\pi\)
							−0.950619	+	0.310361i	\(0.899550\pi\)
\(41\)	42.6584	+	45.9748i	1.04045	+	1.12134i	0.992470	+	0.122488i	\(0.0390872\pi\)
							0.0479777	+	0.998848i	\(0.484722\pi\)
\(43\)	10.7884	−	10.0102i	0.250894	−	0.232795i	−0.544692	−	0.838636i	\(-0.683354\pi\)
							0.795586	+	0.605841i	\(0.207163\pi\)
\(47\)	−0.832549	−	2.12130i	−0.0177138	−	0.0451341i	0.921743	−	0.387801i	\(-0.126765\pi\)
							−0.939457	+	0.342667i	\(0.888670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	211.3.m.a.138.14	yes	408
211.26	odd	42	inner	211.3.m.a.26.14	✓	408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.3.m.a.26.14	✓	408	211.26	odd	42	inner
211.3.m.a.138.14	yes	408	1.1	even	1	trivial