Embedded newform 29.6.e.a.22.10

• Label: 29.6.e.a.22.10
• Level: $29$
• Weight: $6$
• Character: 29.22
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.10
Character	\(\chi\)	\(=\)	29.22
Dual form			29.6.e.a.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.78331 - 1.54825i) q^{2} +(11.8503 - 24.6073i) q^{3} +(14.7853 - 7.12021i) q^{4} +(10.0145 + 43.8764i) q^{5} +(42.2859 - 185.266i) q^{6} +(-23.5325 - 11.3326i) q^{7} +(-84.8041 + 67.6290i) q^{8} +(-313.584 - 393.222i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.78331	−	1.54825i	1.19913	−	0.273694i	0.424100	−	0.905616i	\(-0.360591\pi\)
							0.775032	+	0.631922i	\(0.217734\pi\)
\(3\)	11.8503	−	24.6073i	0.760195	−	1.57856i	−0.0543913	−	0.998520i	\(-0.517322\pi\)
							0.814587	−	0.580042i	\(-0.196964\pi\)
\(5\)	10.0145	+	43.8764i	0.179145	+	0.784886i	0.982026	+	0.188746i	\(0.0604423\pi\)
							−0.802881	+	0.596139i	\(0.796701\pi\)
\(7\)	−23.5325	−	11.3326i	−0.181519	−	0.0874150i	0.340917	−	0.940093i	\(-0.389262\pi\)
							−0.522436	+	0.852678i	\(0.674977\pi\)
\(11\)	509.057	+	405.959i	1.26848	+	1.01158i	0.998817	+	0.0486273i	\(0.0154847\pi\)
							0.269666	+	0.962954i	\(0.413087\pi\)
\(13\)	255.443	−	320.315i	0.419214	−	0.525678i	−0.526719	−	0.850039i	\(-0.676578\pi\)
							0.945933	+	0.324362i	\(0.105150\pi\)
\(17\)			738.263i			0.619568i	0.950807	+	0.309784i	\(0.100257\pi\)
							−0.950807	+	0.309784i	\(0.899743\pi\)
\(19\)	−676.200	−	1404.14i	−0.429725	−	0.892334i	−0.997601	−	0.0692200i	\(-0.977949\pi\)
							0.567876	−	0.823114i	\(-0.307765\pi\)
\(23\)	−498.296	+	2183.18i	−0.196412	+	0.860537i	0.776639	+	0.629946i	\(0.216923\pi\)
							−0.973051	+	0.230591i	\(0.925934\pi\)
\(29\)	−4452.94	−	826.117i	−0.983223	−	0.182409i
							\(31\)	−9138.75	+	2085.86i	−1.70798	+	0.389835i	−0.961334	−	0.275387i	\(-0.911194\pi\)
−0.746646	+	0.665222i	\(0.768337\pi\)
\(37\)	8767.82	−	6992.11i	1.05290	−	0.839661i	0.0654927	−	0.997853i	\(-0.479138\pi\)
							0.987408	+	0.158192i	\(0.0505667\pi\)
\(41\)		−	773.955i		−	0.0719045i	−0.999354	−	0.0359523i	\(-0.988554\pi\)
							0.999354	−	0.0359523i	\(-0.0114464\pi\)
\(43\)	14562.4	+	3323.77i	1.20105	+	0.274132i	0.775820	−	0.630954i	\(-0.217336\pi\)
							0.425230	+	0.905086i	\(0.360193\pi\)
\(47\)	−18942.7	−	15106.3i	−1.25083	−	0.997502i	−0.999562	−	0.0295842i	\(-0.990582\pi\)
							−0.251266	−	0.967918i	\(-0.580847\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.22.10	yes	72
29.2	odd	28		841.6.a.l.1.56		72
29.4	even	14	inner	29.6.e.a.4.10	✓	72
29.27	odd	28		841.6.a.l.1.17		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.4.10	✓	72	29.4	even	14	inner
29.6.e.a.22.10	yes	72	1.1	even	1	trivial
841.6.a.l.1.17		72	29.27	odd	28
841.6.a.l.1.56		72	29.2	odd	28