Embedded newform 29.9.f.a.15.13

• Label: 29.9.f.a.15.13
• Level: $29$
• Weight: $9$
• Character: 29.15
• Analytic conductor: $11.814$
• Analytic rank: $0$
• Dimension: $228$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8139796918\)
Analytic rank:	\(0\)
Dimension:	\(228\)
Relative dimension:	\(19\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			15.13
Character	\(\chi\)	\(=\)	29.15
Dual form			29.9.f.a.2.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.2835 - 1.60937i) q^{2} +(128.283 - 80.6053i) q^{3} +(-48.1524 + 10.9905i) q^{4} +(-467.828 - 373.081i) q^{5} +(1702.60 - 1357.78i) q^{6} +(469.147 - 2055.47i) q^{7} +(-4143.33 + 1449.81i) q^{8} +(7112.49 - 14769.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{27}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.2835	−	1.60937i	0.892721	−	0.100585i	0.346327	−	0.938114i	\(-0.387429\pi\)
							0.546394	+	0.837528i	\(0.316000\pi\)
\(3\)	128.283	−	80.6053i	1.58374	−	0.995127i	0.603483	−	0.797376i	\(-0.293779\pi\)
							0.980252	−	0.197751i	\(-0.0633636\pi\)
\(5\)	−467.828	−	373.081i	−0.748525	−	0.596929i	0.173148	−	0.984896i	\(-0.444606\pi\)
							−0.921673	+	0.387967i	\(0.873177\pi\)
\(7\)	469.147	−	2055.47i	0.195396	−	0.856088i	−0.778237	−	0.627970i	\(-0.783886\pi\)
							0.973634	−	0.228117i	\(-0.0732569\pi\)
\(11\)	3281.42	+	1148.22i	0.224125	+	0.0784248i	0.440006	−	0.897995i	\(-0.354976\pi\)
							−0.215880	+	0.976420i	\(0.569262\pi\)
\(13\)	15354.5	+	31883.9i	0.537602	+	1.11634i	0.976042	+	0.217582i	\(0.0698170\pi\)
							−0.438440	+	0.898761i	\(0.644469\pi\)
\(17\)	58865.3	+	58865.3i	0.704797	+	0.704797i	0.965436	−	0.260639i	\(-0.0839334\pi\)
							−0.260639	+	0.965436i	\(0.583933\pi\)
\(19\)	104673.	−	166586.i	0.803192	−	1.27827i	−0.153784	−	0.988105i	\(-0.549146\pi\)
							0.956976	−	0.290168i	\(-0.0937112\pi\)
\(23\)	105113.	+	131807.i	0.375615	+	0.471007i	0.933327	−	0.359028i	\(-0.116892\pi\)
							−0.557711	+	0.830035i	\(0.688320\pi\)
\(29\)	462133.	+	535424.i	0.653394	+	0.757018i
							\(31\)	114780.	−	12932.5i	0.124285	−	0.0140035i	−0.0496030	−	0.998769i	\(-0.515796\pi\)
0.173888	+	0.984765i	\(0.444367\pi\)
\(37\)	−2.78593e6	+	974840.i	−1.48650	+	0.520147i	−0.946831	−	0.321733i	\(-0.895735\pi\)
							−0.539665	+	0.841880i	\(0.681449\pi\)
\(41\)	1.13415e6	−	1.13415e6i	0.401359	−	0.401359i	−0.477353	−	0.878712i	\(-0.658404\pi\)
							0.878712	+	0.477353i	\(0.158404\pi\)
\(43\)	−456696.	+	4.05329e6i	−0.133584	+	1.18559i	0.730742	+	0.682654i	\(0.239174\pi\)
							−0.864325	+	0.502933i	\(0.832254\pi\)
\(47\)	−314564.	+	898972.i	−0.0644640	+	0.184228i	−0.971610	−	0.236589i	\(-0.923971\pi\)
							0.907146	+	0.420816i	\(0.138256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.15.13	yes	228
29.2	odd	28	inner	29.9.f.a.2.13	✓	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.13	✓	228	29.2	odd	28	inner
29.9.f.a.15.13	yes	228	1.1	even	1	trivial