Embedded newform 29.9.f.a.2.17

• Label: 29.9.f.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	29.2
Dual form			29.9.f.a.15.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(25.7386 + 2.90004i) q^{2} +(82.6049 + 51.9041i) q^{3} +(404.482 + 92.3203i) q^{4} +(418.219 - 333.519i) q^{5} +(1975.61 + 1575.49i) q^{6} +(-324.689 - 1422.56i) q^{7} +(3884.38 + 1359.20i) q^{8} +(1282.82 + 2663.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 228 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 3664 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	25.7386	+	2.90004i	1.60866	+	0.181252i	0.870079	−	0.492912i	\(-0.164068\pi\)
							0.738581	+	0.674165i	\(0.235496\pi\)
\(3\)	82.6049	+	51.9041i	1.01981	+	0.640791i	0.934647	−	0.355576i	\(-0.115715\pi\)
							0.0851657	+	0.996367i	\(0.472858\pi\)
\(5\)	418.219	−	333.519i	0.669151	−	0.533630i	−0.228940	−	0.973441i	\(-0.573526\pi\)
							0.898091	+	0.439811i	\(0.144954\pi\)
\(7\)	−324.689	−	1422.56i	−0.135231	−	0.592484i	−0.996445	−	0.0842429i	\(-0.973153\pi\)
							0.861215	−	0.508242i	\(-0.169704\pi\)
\(11\)	−9553.32	+	3342.85i	−0.652505	+	0.228321i	−0.636174	−	0.771545i	\(-0.719484\pi\)
							−0.0163303	+	0.999867i	\(0.505198\pi\)
\(13\)	−14292.7	+	29679.2i	−0.500428	+	1.03915i	0.485847	+	0.874044i	\(0.338511\pi\)
							−0.986276	+	0.165106i	\(0.947203\pi\)
\(17\)	−29451.1	+	29451.1i	−0.352619	+	0.352619i	−0.861083	−	0.508464i	\(-0.830213\pi\)
							0.508464	+	0.861083i	\(0.330213\pi\)
\(19\)	−21585.7	−	34353.5i	−0.165635	−	0.263607i	0.753585	−	0.657350i	\(-0.228323\pi\)
							−0.919220	+	0.393743i	\(0.871180\pi\)
\(23\)	29611.6	−	37131.8i	0.105816	−	0.132689i	−0.726104	−	0.687585i	\(-0.758671\pi\)
							0.831919	+	0.554896i	\(0.187242\pi\)
\(29\)	666386.	+	237015.i	0.942180	+	0.335107i
							\(31\)	1.34647e6	+	151711.i	1.45798	+	0.164275i	0.805077	−	0.593170i	\(-0.202124\pi\)
0.652899	+	0.757445i	\(0.273552\pi\)
\(37\)	−2.21954e6	−	776649.i	−1.18428	−	0.414398i	−0.334939	−	0.942240i	\(-0.608716\pi\)
							−0.849343	+	0.527841i	\(0.823002\pi\)
\(41\)	−3.59064e6	−	3.59064e6i	−1.27068	−	1.27068i	−0.945731	−	0.324949i	\(-0.894653\pi\)
							−0.324949	−	0.945731i	\(-0.605347\pi\)
\(43\)	80312.5	+	712793.i	0.0234914	+	0.208492i	0.999975	−	0.00710065i	\(-0.00226023\pi\)
							−0.976483	+	0.215593i	\(0.930832\pi\)
\(47\)	−92849.2	−	265348.i	−0.0190277	−	0.0543781i	0.933978	−	0.357331i	\(-0.116313\pi\)
							−0.953006	+	0.302953i	\(0.902027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.f.a.2.17	✓	228
29.15	odd	28	inner	29.9.f.a.15.17	yes	228

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.f.a.2.17	✓	228	1.1	even	1	trivial
29.9.f.a.15.17	yes	228	29.15	odd	28	inner