Embedded newform 211.3.m.a.26.14

• Label: 211.3.m.a.26.14
• Level: $211$
• Weight: $3$
• Character: 211.26
• 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			26.14
Character	\(\chi\)	\(=\)	211.26
Dual form			211.3.m.a.138.14

$q$-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 \(\left(\mathbb{Z}/211\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.3.m.a.26.14	✓	408	1.1	even	1	trivial
211.3.m.a.138.14	yes	408	211.138	odd	42	inner