Embedded newform 429.2.t.a.296.4

• Label: 429.2.t.a.296.4
• 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			296.4
Character	\(\chi\)	\(=\)	429.296
Dual form			429.2.t.a.329.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.17254 + 1.25431i) q^{2} +(-0.817753 - 1.52685i) q^{3} +(2.14661 - 3.71803i) q^{4} -2.90929 q^{5} +(3.69175 + 2.29142i) q^{6} +(-0.464602 + 0.804714i) q^{7} +5.75282i q^{8} +(-1.66256 + 2.49718i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.17254	+	1.25431i	−1.53621	+	0.886934i	−0.537159	+	0.843481i	\(0.680503\pi\)
							−0.999055	+	0.0434530i	\(0.986164\pi\)
\(3\)	−0.817753	−	1.52685i	−0.472130	−	0.881529i
							\(5\)	−2.90929			−1.30107			−0.650536	−	0.759475i	\(-0.725456\pi\)
−0.650536	+	0.759475i	\(0.725456\pi\)
\(7\)	−0.464602	+	0.804714i	−0.175603	+	0.304153i	−0.940370	−	0.340154i	\(-0.889521\pi\)
							0.764767	+	0.644307i	\(0.222854\pi\)
\(11\)	−2.70403	−	1.92048i	−0.815294	−	0.579047i
							\(13\)	−2.51408	−	2.58445i	−0.697280	−	0.716799i
\(17\)	1.57788	−	2.73297i	0.382693	−	0.662843i								−0.608753	−	0.793360i	\(-0.708330\pi\)
							0.991446	+	0.130516i	\(0.0416634\pi\)
\(19\)	−2.89165	+	5.00849i	−0.663391	+	1.14903i	0.316328	+	0.948650i	\(0.397550\pi\)
							−0.979719	+	0.200377i	\(0.935783\pi\)
\(23\)	5.34031	−	3.08323i	1.11353	−	0.642897i	0.173789	−	0.984783i	\(-0.444399\pi\)
							0.939742	+	0.341886i	\(0.111066\pi\)
\(29\)	4.40119	+	7.62308i	0.817280	+	1.41557i	0.907679	+	0.419665i	\(0.137852\pi\)
							−0.0903989	+	0.995906i	\(0.528814\pi\)
\(31\)			1.61879i			0.290744i	0.989377	+	0.145372i	\(0.0464379\pi\)
							−0.989377	+	0.145372i	\(0.953562\pi\)
\(37\)	4.77627	−	2.75758i	0.785214	−	0.453344i	−0.0530608	−	0.998591i	\(-0.516898\pi\)
							0.838275	+	0.545248i	\(0.183564\pi\)
\(41\)	1.92098	−	1.10908i	0.300007	−	0.173209i	−0.342439	−	0.939540i	\(-0.611253\pi\)
							0.642446	+	0.766331i	\(0.277920\pi\)
\(43\)	1.86427	+	1.07634i	0.284299	+	0.164140i	0.635368	−	0.772209i	\(-0.280848\pi\)
							−0.351069	+	0.936350i	\(0.614182\pi\)
\(47\)	−1.23936			−0.180779			−0.0903894	−	0.995907i	\(-0.528811\pi\)
							−0.0903894	+	0.995907i	\(0.528811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.296.4	yes	104
3.2	odd	2	inner	429.2.t.a.296.49	yes	104
11.10	odd	2	inner	429.2.t.a.296.50	yes	104
13.4	even	6	inner	429.2.t.a.329.3	yes	104
33.32	even	2	inner	429.2.t.a.296.3	✓	104
39.17	odd	6	inner	429.2.t.a.329.50	yes	104
143.43	odd	6	inner	429.2.t.a.329.49	yes	104
429.329	even	6	inner	429.2.t.a.329.4	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.3	✓	104	33.32	even	2	inner
429.2.t.a.296.4	yes	104	1.1	even	1	trivial
429.2.t.a.296.49	yes	104	3.2	odd	2	inner
429.2.t.a.296.50	yes	104	11.10	odd	2	inner
429.2.t.a.329.3	yes	104	13.4	even	6	inner
429.2.t.a.329.4	yes	104	429.329	even	6	inner
429.2.t.a.329.49	yes	104	143.43	odd	6	inner
429.2.t.a.329.50	yes	104	39.17	odd	6	inner