Embedded newform 429.2.bg.b.16.5

• Label: 429.2.bg.b.16.5
• 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			16.5
Character	\(\chi\)	\(=\)	429.16
Dual form			429.2.bg.b.295.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.802761 + 0.891557i) q^{2} +(-0.104528 + 0.994522i) q^{3} +(0.0586092 + 0.557629i) q^{4} +(0.833954 - 2.56665i) q^{5} +(-0.802761 - 0.891557i) q^{6} +(0.259398 + 2.46800i) q^{7} +(-2.48538 - 1.80573i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 14 q^{3} + 8 q^{4} - 6 q^{5} - 24 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.802761	+	0.891557i	−0.567638	+	0.630426i	−0.956801	−	0.290743i	\(-0.906098\pi\)
							0.389163	+	0.921169i	\(0.372764\pi\)
\(3\)	−0.104528	+	0.994522i	−0.0603495	+	0.574187i
							\(5\)	0.833954	−	2.56665i	0.372956	−	1.14784i	−0.571892	−	0.820329i	\(-0.693790\pi\)
0.944848	−	0.327510i	\(-0.106210\pi\)
\(7\)	0.259398	+	2.46800i	0.0980431	+	0.932818i	0.927393	+	0.374089i	\(0.122044\pi\)
							−0.829350	+	0.558730i	\(0.811289\pi\)
\(11\)	−3.17436	+	0.960959i	−0.957105	+	0.289740i
							\(13\)	−2.81603	+	2.25166i	−0.781027	+	0.624498i
\(17\)	2.79265	+	3.10155i	0.677317	+	0.752237i								0.979595	−	0.200983i	\(-0.0644136\pi\)
							−0.302278	+	0.953220i	\(0.597747\pi\)
\(19\)	−6.39200	+	2.84590i	−1.46643	+	0.652895i	−0.975838	−	0.218497i	\(-0.929885\pi\)
							−0.490588	+	0.871392i	\(0.663218\pi\)
\(23\)	2.37560	+	4.11465i	0.495346	+	0.857964i	0.999986	−	0.00536566i	\(-0.00170795\pi\)
							−0.504640	+	0.863330i	\(0.668375\pi\)
\(29\)	−4.97554	−	2.21525i	−0.923935	−	0.411362i	−0.111069	−	0.993813i	\(-0.535427\pi\)
							−0.812866	+	0.582450i	\(0.802094\pi\)
\(31\)	1.37791	+	4.24077i	0.247480	+	0.761665i	0.995219	+	0.0976722i	\(0.0311397\pi\)
							−0.747739	+	0.663993i	\(0.768860\pi\)
\(37\)	−9.17224	−	4.08374i	−1.50791	−	0.671363i	−0.524274	−	0.851550i	\(-0.675663\pi\)
							−0.983632	+	0.180187i	\(0.942330\pi\)
\(41\)	−0.491734	+	4.67854i	−0.0767959	+	0.730665i	0.886592	+	0.462553i	\(0.153066\pi\)
							−0.963388	+	0.268112i	\(0.913600\pi\)
\(43\)	5.40742	−	9.36593i	0.824624	−	1.42829i	−0.0775819	−	0.996986i	\(-0.524720\pi\)
							0.902206	−	0.431305i	\(-0.141947\pi\)
\(47\)	4.10011	+	2.97890i	0.598062	+	0.434518i	0.845191	−	0.534465i	\(-0.179487\pi\)
							−0.247128	+	0.968983i	\(0.579487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bg.b.16.5	✓	112
11.9	even	5	inner	429.2.bg.b.328.10	yes	112
13.9	even	3	inner	429.2.bg.b.412.10	yes	112
143.9	even	15	inner	429.2.bg.b.295.5	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bg.b.16.5	✓	112	1.1	even	1	trivial
429.2.bg.b.295.5	yes	112	143.9	even	15	inner
429.2.bg.b.328.10	yes	112	11.9	even	5	inner
429.2.bg.b.412.10	yes	112	13.9	even	3	inner