Embedded newform 429.2.bg.b.16.9

• Label: 429.2.bg.b.16.9
• Level: $429$
• Weight: $2$
• Character: 429.16
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bg (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.9
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.b.295.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.433437 - 0.481381i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.165197 + 1.57175i) q^{4} +(-0.543246 + 1.67194i) q^{5} +(0.433437 + 0.481381i) q^{6} +(0.0903479 + 0.859603i) q^{7} +(1.87631 + 1.36322i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.433437	−	0.481381i	0.306486	−	0.340388i	−0.570150	−	0.821540i	\(-0.693115\pi\)
							0.876637	+	0.481153i	\(0.159782\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	−0.543246	+	1.67194i	−0.242947	+	0.747714i	0.753020	+	0.657997i	\(0.228596\pi\)
−0.995967	+	0.0897169i	\(0.971404\pi\)
\(7\)	0.0903479	+	0.859603i	0.0341483	+	0.324899i	0.998239	+	0.0593251i	\(0.0188948\pi\)
							−0.964090	+	0.265574i	\(0.914438\pi\)
\(11\)	−3.16725	+	0.984153i	−0.954960	+	0.296733i
							\(13\)	−3.15105	−	1.75240i	−0.873943	−	0.486028i
\(17\)	−3.67181	−	4.07795i	−0.890544	−	0.989049i								0.109444	−	0.993993i	\(-0.465093\pi\)
							−0.999988	+	0.00494413i	\(0.998426\pi\)
\(19\)	5.58465	−	2.48644i	1.28121	−	0.570429i	0.350625	−	0.936516i	\(-0.385969\pi\)
							0.930581	+	0.366087i	\(0.119303\pi\)
\(23\)	4.13552	+	7.16293i	0.862315	+	1.49357i	0.869689	+	0.493600i	\(0.164319\pi\)
							−0.00737395	+	0.999973i	\(0.502347\pi\)
\(29\)	−1.50116	−	0.668359i	−0.278758	−	0.124111i	0.262596	−	0.964906i	\(-0.415421\pi\)
							−0.541354	+	0.840795i	\(0.682088\pi\)
\(31\)	2.00146	+	6.15987i	0.359473	+	1.10634i	0.953370	+	0.301804i	\(0.0975887\pi\)
							−0.593897	+	0.804541i	\(0.702411\pi\)
\(37\)	6.43424	+	2.86471i	1.05778	+	0.470955i	0.860531	−	0.509398i	\(-0.170132\pi\)
							0.197251	+	0.980353i	\(0.436799\pi\)
\(41\)	−0.0687854	+	0.654449i	−0.0107425	+	0.102208i	−0.998579	−	0.0532865i	\(-0.983030\pi\)
							0.987837	+	0.155494i	\(0.0496970\pi\)
\(43\)	1.03849	−	1.79872i	0.158368	−	0.274302i	−0.775912	−	0.630841i	\(-0.782710\pi\)
							0.934280	+	0.356539i	\(0.116043\pi\)
\(47\)	−1.01685	−	0.738787i	−0.148323	−	0.107763i	0.511149	−	0.859492i	\(-0.329220\pi\)
							−0.659472	+	0.751729i	\(0.729220\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.16.9	✓	112
11.9	even	5	inner	429.2.bg.b.328.6	yes	112
13.9	even	3	inner	429.2.bg.b.412.6	yes	112
143.9	even	15	inner	429.2.bg.b.295.9	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.16.9	✓	112	1.1	even	1	trivial
429.2.bg.b.295.9	yes	112	143.9	even	15	inner
429.2.bg.b.328.6	yes	112	11.9	even	5	inner
429.2.bg.b.412.6	yes	112	13.9	even	3	inner