Embedded newform 429.2.t.a.296.8

• Label: 429.2.t.a.296.8
• 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			296.8
Character	\(\chi\)	\(=\)	429.296
Dual form			429.2.t.a.329.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.92182 + 1.10956i) q^{2} +(1.37961 - 1.04722i) q^{3} +(1.46227 - 2.53272i) q^{4} -1.97176 q^{5} +(-1.48942 + 3.54334i) q^{6} +(-0.477966 + 0.827861i) q^{7} +2.05166i q^{8} +(0.806667 - 2.88951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.92182	+	1.10956i	−1.35893	+	0.784581i	−0.989480	−	0.144669i	\(-0.953788\pi\)
							−0.369453	+	0.929249i	\(0.620455\pi\)
\(3\)	1.37961	−	1.04722i	0.796520	−	0.604612i
							\(5\)	−1.97176			−0.881797			−0.440899	−	0.897557i	\(-0.645340\pi\)
−0.440899	+	0.897557i	\(0.645340\pi\)
\(7\)	−0.477966	+	0.827861i	−0.180654	+	0.312902i	−0.942103	−	0.335322i	\(-0.891155\pi\)
							0.761449	+	0.648224i	\(0.224488\pi\)
\(11\)	−1.60105	+	2.90459i	−0.482734	+	0.875767i
							\(13\)	3.43390	−	1.09923i	0.952394	−	0.304870i
\(17\)	1.09581	−	1.89800i	0.265773	−	0.460331i								−0.701993	−	0.712184i	\(-0.747706\pi\)
							0.967766	+	0.251852i	\(0.0810397\pi\)
\(19\)	2.70718	−	4.68897i	0.621069	−	1.07572i	−0.368218	−	0.929740i	\(-0.620032\pi\)
							0.989287	−	0.145984i	\(-0.0466348\pi\)
\(23\)	5.88960	−	3.40036i	1.22807	−	0.709024i	0.261441	−	0.965220i	\(-0.415802\pi\)
							0.966625	+	0.256196i	\(0.0824691\pi\)
\(29\)	−3.45995	−	5.99282i	−0.642497	−	1.11284i	−0.984874	−	0.173275i	\(-0.944565\pi\)
							0.342376	−	0.939563i	\(-0.388768\pi\)
\(31\)		−	9.45661i		−	1.69846i	−0.528025	−	0.849229i	\(-0.677067\pi\)
							0.528025	−	0.849229i	\(-0.322933\pi\)
\(37\)	−3.85093	+	2.22334i	−0.633090	+	0.365515i	−0.781948	−	0.623344i	\(-0.785774\pi\)
							0.148858	+	0.988859i	\(0.452440\pi\)
\(41\)	7.34788	−	4.24230i	1.14755	−	0.662536i	0.199258	−	0.979947i	\(-0.436147\pi\)
							0.948288	+	0.317411i	\(0.102814\pi\)
\(43\)	−3.90995	−	2.25741i	−0.596261	−	0.344252i	0.171308	−	0.985218i	\(-0.445201\pi\)
							−0.767569	+	0.640966i	\(0.778534\pi\)
\(47\)	8.69313			1.26802			0.634012	−	0.773323i	\(-0.281407\pi\)
							0.634012	+	0.773323i	\(0.281407\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.296.8	yes	104
3.2	odd	2	inner	429.2.t.a.296.45	yes	104
11.10	odd	2	inner	429.2.t.a.296.46	yes	104
13.4	even	6	inner	429.2.t.a.329.7	yes	104
33.32	even	2	inner	429.2.t.a.296.7	✓	104
39.17	odd	6	inner	429.2.t.a.329.46	yes	104
143.43	odd	6	inner	429.2.t.a.329.45	yes	104
429.329	even	6	inner	429.2.t.a.329.8	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.7	✓	104	33.32	even	2	inner
429.2.t.a.296.8	yes	104	1.1	even	1	trivial
429.2.t.a.296.45	yes	104	3.2	odd	2	inner
429.2.t.a.296.46	yes	104	11.10	odd	2	inner
429.2.t.a.329.7	yes	104	13.4	even	6	inner
429.2.t.a.329.8	yes	104	429.329	even	6	inner
429.2.t.a.329.45	yes	104	143.43	odd	6	inner
429.2.t.a.329.46	yes	104	39.17	odd	6	inner