Embedded newform 429.2.t.a.296.1

• Label: 429.2.t.a.296.1
• Level: $429$
• Weight: $2$
• Character: 429.296
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(52\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			296.1
Character	\(\chi\)	\(=\)	429.296
Dual form			429.2.t.a.329.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.31547 + 1.33684i) q^{2} +(0.0244717 + 1.73188i) q^{3} +(2.57426 - 4.45876i) q^{4} +2.76446 q^{5} +(-2.37190 - 3.97739i) q^{6} +(0.565134 - 0.978841i) q^{7} +8.41814i q^{8} +(-2.99880 + 0.0847639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.31547	+	1.33684i	−1.63728	+	0.945286i	−0.655521	+	0.755177i	\(0.727551\pi\)
							−0.981763	+	0.190109i	\(0.939116\pi\)
\(3\)	0.0244717	+	1.73188i	0.0141287	+	0.999900i
							\(5\)	2.76446			1.23630			0.618152	−	0.786059i	\(-0.287882\pi\)
0.618152	+	0.786059i	\(0.287882\pi\)
\(7\)	0.565134	−	0.978841i	0.213601	−	0.369967i	−0.739238	−	0.673444i	\(-0.764814\pi\)
							0.952839	+	0.303477i	\(0.0981475\pi\)
\(11\)	0.738411	−	3.23338i	0.222639	−	0.974901i
							\(13\)	2.86710	−	2.18626i	0.795190	−	0.606361i
\(17\)	1.85321	−	3.20985i	0.449469	−	0.778504i								−0.548882	−	0.835900i	\(-0.684946\pi\)
							0.998351	+	0.0573959i	\(0.0182797\pi\)
\(19\)	0.848514	−	1.46967i	0.194662	−	0.337165i	−0.752127	−	0.659018i	\(-0.770972\pi\)
							0.946790	+	0.321853i	\(0.104306\pi\)
\(23\)	7.29596	−	4.21232i	1.52131	−	0.878330i	0.521628	−	0.853173i	\(-0.325325\pi\)
							0.999684	−	0.0251570i	\(-0.00800856\pi\)
\(29\)	−0.295904	−	0.512520i	−0.0549479	−	0.0951726i	0.837243	−	0.546831i	\(-0.184166\pi\)
							−0.892191	+	0.451658i	\(0.850833\pi\)
\(31\)		−	4.08080i		−	0.732933i	−0.930431	−	0.366467i	\(-0.880567\pi\)
							0.930431	−	0.366467i	\(-0.119433\pi\)
\(37\)	−8.55581	+	4.93970i	−1.40657	+	0.812082i	−0.995055	−	0.0993227i	\(-0.968332\pi\)
							−0.411512	+	0.911404i	\(0.634999\pi\)
\(41\)	−8.76317	+	5.05942i	−1.36858	+	0.790148i	−0.990746	−	0.135726i	\(-0.956663\pi\)
							−0.377831	+	0.925875i	\(0.623330\pi\)
\(43\)	7.31991	+	4.22615i	1.11628	+	0.644482i	0.940448	−	0.339939i	\(-0.110406\pi\)
							0.175828	+	0.984421i	\(0.443740\pi\)
\(47\)	1.04557			0.152512			0.0762559	−	0.997088i	\(-0.475703\pi\)
							0.0762559	+	0.997088i	\(0.475703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.296.1	✓	104
3.2	odd	2	inner	429.2.t.a.296.52	yes	104
11.10	odd	2	inner	429.2.t.a.296.51	yes	104
13.4	even	6	inner	429.2.t.a.329.2	yes	104
33.32	even	2	inner	429.2.t.a.296.2	yes	104
39.17	odd	6	inner	429.2.t.a.329.51	yes	104
143.43	odd	6	inner	429.2.t.a.329.52	yes	104
429.329	even	6	inner	429.2.t.a.329.1	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.1	✓	104	1.1	even	1	trivial
429.2.t.a.296.2	yes	104	33.32	even	2	inner
429.2.t.a.296.51	yes	104	11.10	odd	2	inner
429.2.t.a.296.52	yes	104	3.2	odd	2	inner
429.2.t.a.329.1	yes	104	429.329	even	6	inner
429.2.t.a.329.2	yes	104	13.4	even	6	inner
429.2.t.a.329.51	yes	104	39.17	odd	6	inner
429.2.t.a.329.52	yes	104	143.43	odd	6	inner