Embedded newform 29.9.f.a.2.6

• Label: 29.9.f.a.2.6
• Level: $29$
• Weight: $9$
• Character: 29.2
• 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			2.6
Character	\(\chi\)	\(=\)	29.2
Dual form			29.9.f.a.15.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.5588 - 2.09108i) q^{2} +(-58.4072 - 36.6997i) q^{3} +(90.4766 + 20.6507i) q^{4} +(495.290 - 394.980i) q^{5} +(1007.23 + 803.238i) q^{6} +(641.791 + 2811.87i) q^{7} +(2876.87 + 1006.66i) q^{8} +(-782.176 - 1624.21i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.5588	−	2.09108i	−1.15993	−	0.130692i	−0.489047	−	0.872258i	\(-0.662655\pi\)
							−0.670881	+	0.741565i	\(0.734084\pi\)
\(3\)	−58.4072	−	36.6997i	−0.721077	−	0.453082i	0.120853	−	0.992670i	\(-0.461437\pi\)
							−0.841929	+	0.539588i	\(0.818580\pi\)
\(5\)	495.290	−	394.980i	0.792464	−	0.631969i	−0.141256	−	0.989973i	\(-0.545114\pi\)
							0.933720	+	0.358004i	\(0.116543\pi\)
\(7\)	641.791	+	2811.87i	0.267302	+	1.17113i	0.913138	+	0.407650i	\(0.133652\pi\)
							−0.645837	+	0.763476i	\(0.723491\pi\)
\(11\)	14471.5	−	5063.78i	0.988420	−	0.345863i	0.212822	−	0.977091i	\(-0.431734\pi\)
							0.775597	+	0.631228i	\(0.217449\pi\)
\(13\)	1393.52	−	2893.67i	0.0487909	−	0.101315i	−0.875148	−	0.483855i	\(-0.839236\pi\)
							0.923939	+	0.382540i	\(0.124950\pi\)
\(17\)	−14700.9	+	14700.9i	−0.176014	+	0.176014i	−0.789616	−	0.613601i	\(-0.789720\pi\)
							0.613601	+	0.789616i	\(0.289720\pi\)
\(19\)	−100542.	−	160012.i	−0.771497	−	1.22783i	−0.968887	−	0.247505i	\(-0.920389\pi\)
							0.197390	−	0.980325i	\(-0.436754\pi\)
\(23\)	205776.	−	258035.i	0.735332	−	0.922077i	−0.263764	−	0.964587i	\(-0.584964\pi\)
							0.999096	+	0.0425105i	\(0.0135356\pi\)
\(29\)	−702763.	+	79815.0i	−0.993612	+	0.112848i
							\(31\)	509594.	+	57417.5i	0.551795	+	0.0621724i	0.383460	−	0.923557i	\(-0.374732\pi\)
0.168335	+	0.985730i	\(0.446161\pi\)
\(37\)	792841.	+	277427.i	0.423038	+	0.148027i	0.533398	−	0.845864i	\(-0.320915\pi\)
							−0.110360	+	0.993892i	\(0.535201\pi\)
\(41\)	−1.60774e6	−	1.60774e6i	−0.568957	−	0.568957i	0.362880	−	0.931836i	\(-0.381794\pi\)
							−0.931836	+	0.362880i	\(0.881794\pi\)
\(43\)	−698958.	−	6.20342e6i	−0.204445	−	1.81450i	−0.507634	−	0.861573i	\(-0.669480\pi\)
							0.303189	−	0.952930i	\(-0.401949\pi\)
\(47\)	−1.56085e6	−	4.46066e6i	−0.319867	−	0.914129i	−0.985771	−	0.168091i	\(-0.946240\pi\)
							0.665904	−	0.746037i	\(-0.268046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.2.6	✓	228
29.15	odd	28	inner	29.9.f.a.15.6	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.6	✓	228	1.1	even	1	trivial
29.9.f.a.15.6	yes	228	29.15	odd	28	inner