Embedded newform 429.2.t.a.296.14

• Label: 429.2.t.a.296.14
• 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			296.14
Character	\(\chi\)	\(=\)	429.296
Dual form			429.2.t.a.329.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25571 + 0.724987i) q^{2} +(0.108816 - 1.72863i) q^{3} +(0.0512119 - 0.0887017i) q^{4} +0.906362 q^{5} +(1.11659 + 2.24955i) q^{6} +(-1.12801 + 1.95377i) q^{7} -2.75144i q^{8} +(-2.97632 - 0.376204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25571	+	0.724987i	−0.887924	+	0.512643i	−0.873263	−	0.487249i	\(-0.838000\pi\)
							−0.0146611	+	0.999893i	\(0.504667\pi\)
\(3\)	0.108816	−	1.72863i	0.0628247	−	0.998025i
							\(5\)	0.906362			0.405337			0.202669	−	0.979247i	\(-0.435039\pi\)
0.202669	+	0.979247i	\(0.435039\pi\)
\(7\)	−1.12801	+	1.95377i	−0.426347	+	0.738454i	−0.996545	−	0.0830528i	\(-0.973533\pi\)
							0.570198	+	0.821507i	\(0.306866\pi\)
\(11\)	−3.11407	−	1.14130i	−0.938928	−	0.344114i
							\(13\)	−1.42900	+	3.31028i	−0.396335	+	0.918106i
\(17\)	−1.20981	+	2.09545i	−0.293421	+	0.508221i								−0.974616	−	0.223881i	\(-0.928127\pi\)
							0.681195	+	0.732102i	\(0.261461\pi\)
\(19\)	1.13994	−	1.97444i	0.261521	−	0.452967i	−0.705126	−	0.709082i	\(-0.749109\pi\)
							0.966646	+	0.256115i	\(0.0824427\pi\)
\(23\)	0.733503	−	0.423488i	0.152946	−	0.0883034i	−0.421574	−	0.906794i	\(-0.638522\pi\)
							0.574520	+	0.818491i	\(0.305189\pi\)
\(29\)	−2.92126	−	5.05977i	−0.542465	−	0.939577i	−0.998762	−	0.0497490i	\(-0.984158\pi\)
							0.456297	−	0.889828i	\(-0.349175\pi\)
\(31\)			8.86085i			1.59146i	0.605655	+	0.795728i	\(0.292911\pi\)
							−0.605655	+	0.795728i	\(0.707089\pi\)
\(37\)	−8.58723	+	4.95784i	−1.41173	+	0.815064i	−0.995552	−	0.0942179i	\(-0.969965\pi\)
							−0.416181	+	0.909282i	\(0.636632\pi\)
\(41\)	−9.60348	+	5.54457i	−1.49981	+	0.865917i	−1.00000		0.000217338i	\(-0.999931\pi\)
							−0.499812	+	0.866134i	\(0.666597\pi\)
\(43\)	−5.23210	−	3.02075i	−0.797888	−	0.460661i	0.0448444	−	0.998994i	\(-0.485721\pi\)
							−0.842732	+	0.538333i	\(0.819054\pi\)
\(47\)	5.15850			0.752445			0.376223	−	0.926529i	\(-0.377223\pi\)
							0.376223	+	0.926529i	\(0.377223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.296.14	yes	104
3.2	odd	2	inner	429.2.t.a.296.39	yes	104
11.10	odd	2	inner	429.2.t.a.296.40	yes	104
13.4	even	6	inner	429.2.t.a.329.13	yes	104
33.32	even	2	inner	429.2.t.a.296.13	✓	104
39.17	odd	6	inner	429.2.t.a.329.40	yes	104
143.43	odd	6	inner	429.2.t.a.329.39	yes	104
429.329	even	6	inner	429.2.t.a.329.14	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.13	✓	104	33.32	even	2	inner
429.2.t.a.296.14	yes	104	1.1	even	1	trivial
429.2.t.a.296.39	yes	104	3.2	odd	2	inner
429.2.t.a.296.40	yes	104	11.10	odd	2	inner
429.2.t.a.329.13	yes	104	13.4	even	6	inner
429.2.t.a.329.14	yes	104	429.329	even	6	inner
429.2.t.a.329.39	yes	104	143.43	odd	6	inner
429.2.t.a.329.40	yes	104	39.17	odd	6	inner