Embedded newform 429.2.t.a.329.19

• Label: 429.2.t.a.329.19
• Level: $429$
• Weight: $2$
• Character: 429.329
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $104$
• 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			329.19
Character	\(\chi\)	\(=\)	429.329
Dual form			429.2.t.a.296.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.716153 - 0.413471i) q^{2} +(-1.68064 - 0.418881i) q^{3} +(-0.658083 - 1.13983i) q^{4} -3.60945 q^{5} +(1.03040 + 0.994878i) q^{6} +(-1.45351 - 2.51755i) q^{7} +2.74228i q^{8} +(2.64908 + 1.40797i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 2 q^{3} + 44 q^{4} - 6 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.716153	−	0.413471i	−0.506397	−	0.292368i	0.224955	−	0.974369i	\(-0.427777\pi\)
							−0.731351	+	0.682001i	\(0.761110\pi\)
\(3\)	−1.68064	−	0.418881i	−0.970316	−	0.241841i
							\(5\)	−3.60945			−1.61420			−0.807099	−	0.590417i	\(-0.798963\pi\)
−0.807099	+	0.590417i	\(0.798963\pi\)
\(7\)	−1.45351	−	2.51755i	−0.549375	−	0.951545i	−0.998317	−	0.0579844i	\(-0.981533\pi\)
							0.448943	−	0.893561i	\(-0.351801\pi\)
\(11\)	−2.69945	−	1.92691i	−0.813915	−	0.580984i
							\(13\)	−2.11573	−	2.91953i	−0.586799	−	0.809733i
\(17\)	−0.309839	−	0.536657i	−0.0751470	−	0.130158i								0.826003	−	0.563665i	\(-0.190609\pi\)
							−0.901150	+	0.433507i	\(0.857276\pi\)
\(19\)	1.18049	+	2.04467i	0.270823	+	0.469079i	0.969073	−	0.246775i	\(-0.0793710\pi\)
							−0.698250	+	0.715854i	\(0.746038\pi\)
\(23\)	5.10705	+	2.94856i	1.06489	+	0.614817i	0.926782	−	0.375600i	\(-0.122563\pi\)
							0.138112	+	0.990417i	\(0.455897\pi\)
\(29\)	3.36278	−	5.82450i	0.624452	−	1.08158i	−0.364195	−	0.931323i	\(-0.618656\pi\)
							0.988647	−	0.150259i	\(-0.0480109\pi\)
\(31\)			5.67006i			1.01837i	0.860656	+	0.509186i	\(0.170054\pi\)
							−0.860656	+	0.509186i	\(0.829946\pi\)
\(37\)	0.225314	+	0.130085i	0.0370414	+	0.0213859i	0.518406	−	0.855134i	\(-0.326525\pi\)
							−0.481365	+	0.876520i	\(0.659859\pi\)
\(41\)	−2.92623	−	1.68946i	−0.457000	−	0.263849i	0.253782	−	0.967261i	\(-0.418325\pi\)
							−0.710782	+	0.703412i	\(0.751659\pi\)
\(43\)	−8.07839	+	4.66406i	−1.23194	+	0.711263i	−0.967435	−	0.253121i	\(-0.918543\pi\)
							−0.264508	+	0.964383i	\(0.585210\pi\)
\(47\)	−6.20944			−0.905740			−0.452870	−	0.891577i	\(-0.649600\pi\)
							−0.452870	+	0.891577i	\(0.649600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.t.a.329.19	yes	104
3.2	odd	2	inner	429.2.t.a.329.34	yes	104
11.10	odd	2	inner	429.2.t.a.329.33	yes	104
13.10	even	6	inner	429.2.t.a.296.20	yes	104
33.32	even	2	inner	429.2.t.a.329.20	yes	104
39.23	odd	6	inner	429.2.t.a.296.33	yes	104
143.10	odd	6	inner	429.2.t.a.296.34	yes	104
429.296	even	6	inner	429.2.t.a.296.19	✓	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.t.a.296.19	✓	104	429.296	even	6	inner
429.2.t.a.296.20	yes	104	13.10	even	6	inner
429.2.t.a.296.33	yes	104	39.23	odd	6	inner
429.2.t.a.296.34	yes	104	143.10	odd	6	inner
429.2.t.a.329.19	yes	104	1.1	even	1	trivial
429.2.t.a.329.20	yes	104	33.32	even	2	inner
429.2.t.a.329.33	yes	104	11.10	odd	2	inner
429.2.t.a.329.34	yes	104	3.2	odd	2	inner