Embedded newform 29.9.f.a.2.8

• Label: 29.9.f.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	29.2
Dual form			29.9.f.a.15.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.51607 - 0.621512i) q^{2} +(67.5754 + 42.4604i) q^{3} +(-219.541 - 50.1088i) q^{4} +(264.780 - 211.155i) q^{5} +(-346.361 - 276.213i) q^{6} +(606.153 + 2655.73i) q^{7} +(2521.16 + 882.193i) q^{8} +(-83.1675 - 172.699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.51607	−	0.621512i	−0.344754	−	0.0388445i	−0.0621101	−	0.998069i	\(-0.519783\pi\)
							−0.282644	+	0.959225i	\(0.591212\pi\)
\(3\)	67.5754	+	42.4604i	0.834264	+	0.524203i	0.880123	−	0.474745i	\(-0.157460\pi\)
							−0.0458594	+	0.998948i	\(0.514603\pi\)
\(5\)	264.780	−	211.155i	0.423648	−	0.337848i	−0.388346	−	0.921514i	\(-0.626953\pi\)
							0.811994	+	0.583666i	\(0.198382\pi\)
\(7\)	606.153	+	2655.73i	0.252459	+	1.10609i	0.929114	+	0.369793i	\(0.120572\pi\)
							−0.676655	+	0.736300i	\(0.736571\pi\)
\(11\)	−1969.45	+	689.141i	−0.134516	+	0.0470692i	−0.396698	−	0.917949i	\(-0.629844\pi\)
							0.262181	+	0.965019i	\(0.415558\pi\)
\(13\)	−14575.2	+	30265.7i	−0.510318	+	1.05969i	0.473546	+	0.880769i	\(0.342974\pi\)
							−0.983864	+	0.178917i	\(0.942741\pi\)
\(17\)	−83330.5	+	83330.5i	−0.997719	+	0.997719i	−0.999997	−	0.00227856i	\(-0.999275\pi\)
							0.00227856	+	0.999997i	\(0.499275\pi\)
\(19\)	97302.1	+	154855.i	0.746634	+	1.18826i	0.976600	+	0.215063i	\(0.0689957\pi\)
							−0.229966	+	0.973199i	\(0.573861\pi\)
\(23\)	−175466.	+	220028.i	−0.627022	+	0.786260i	−0.989313	−	0.145805i	\(-0.953423\pi\)
							0.362292	+	0.932065i	\(0.381994\pi\)
\(29\)	389422.	−	590421.i	0.550591	−	0.834775i
							\(31\)	−407677.	−	45934.1i	−0.441437	−	0.0497380i	−0.111551	−	0.993759i	\(-0.535582\pi\)
−0.329886	+	0.944021i	\(0.607010\pi\)
\(37\)	2.81148e6	+	983781.i	1.50013	+	0.524918i	0.950563	−	0.310532i	\(-0.100507\pi\)
							0.549566	+	0.835450i	\(0.314793\pi\)
\(41\)	−355267.	−	355267.i	−0.125724	−	0.125724i	0.641445	−	0.767169i	\(-0.278335\pi\)
							−0.767169	+	0.641445i	\(0.778335\pi\)
\(43\)	−457817.	−	4.06324e6i	−0.133912	−	1.18850i	−0.863406	−	0.504511i	\(-0.831673\pi\)
							0.729494	−	0.683987i	\(-0.239756\pi\)
\(47\)	−1.42653e6	−	4.07679e6i	−0.292341	−	0.835462i	−0.992584	−	0.121558i	\(-0.961211\pi\)
							0.700244	−	0.713904i	\(-0.253075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.2.8	✓	228
29.15	odd	28	inner	29.9.f.a.15.8	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.8	✓	228	1.1	even	1	trivial
29.9.f.a.15.8	yes	228	29.15	odd	28	inner