Embedded newform 29.9.c.a.12.8

• Label: 29.9.c.a.12.8
• Level: $29$
• Weight: $9$
• Character: 29.12
• Analytic conductor: $11.814$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8139796918\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.8
Character	\(\chi\)	\(=\)	29.12
Dual form			29.9.c.a.17.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.50125 - 6.50125i) q^{2} +(25.7599 + 25.7599i) q^{3} -171.468i q^{4} +37.7478i q^{5} -334.943i q^{6} +1420.11 q^{7} +(-2779.07 + 2779.07i) q^{8} -5233.85i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 8 q^{2} - 2 q^{3} - 4 q^{7} - 3678 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.50125	−	6.50125i	−0.406328	−	0.406328i	0.474128	−	0.880456i	\(-0.342763\pi\)
							−0.880456	+	0.474128i	\(0.842763\pi\)
\(3\)	25.7599	+	25.7599i	0.318024	+	0.318024i	0.848008	−	0.529984i	\(-0.177802\pi\)
							−0.529984	+	0.848008i	\(0.677802\pi\)
\(5\)			37.7478i			0.0603964i	0.999544	+	0.0301982i	\(0.00961385\pi\)
							−0.999544	+	0.0301982i	\(0.990386\pi\)
\(7\)	1420.11			0.591465			0.295732	−	0.955271i	\(-0.404436\pi\)
							0.295732	+	0.955271i	\(0.404436\pi\)
\(11\)	1721.35	+	1721.35i	0.117570	+	0.117570i	0.763444	−	0.645874i	\(-0.223507\pi\)
							−0.645874	+	0.763444i	\(0.723507\pi\)
\(13\)		−	13042.7i		−	0.456663i	−0.973583	−	0.228331i	\(-0.926673\pi\)
							0.973583	−	0.228331i	\(-0.0733270\pi\)
\(17\)	−106478.	−	106478.i	−1.27487	−	1.27487i	−0.943503	−	0.331365i	\(-0.892491\pi\)
							−0.331365	−	0.943503i	\(-0.607509\pi\)
\(19\)	−109384.	−	109384.i	−0.839344	−	0.839344i	0.149428	−	0.988773i	\(-0.452257\pi\)
							−0.988773	+	0.149428i	\(0.952257\pi\)
\(23\)	−534222.			−1.90902			−0.954510	−	0.298180i	\(-0.903620\pi\)
							−0.954510	+	0.298180i	\(0.903620\pi\)
\(29\)	595131.	−	382185.i	0.841435	−	0.540358i
							\(31\)	295823.	+	295823.i	0.320321	+	0.320321i	0.848890	−	0.528569i	\(-0.177271\pi\)
−0.528569	+	0.848890i	\(0.677271\pi\)
\(37\)	517328.	−	517328.i	0.276032	−	0.276032i	−0.555491	−	0.831523i	\(-0.687470\pi\)
							0.831523	+	0.555491i	\(0.187470\pi\)
\(41\)	−662466.	+	662466.i	−0.234438	+	0.234438i	−0.814542	−	0.580104i	\(-0.803012\pi\)
							0.580104	+	0.814542i	\(0.303012\pi\)
\(43\)	1.36965e6	+	1.36965e6i	0.400623	+	0.400623i	0.878453	−	0.477829i	\(-0.158576\pi\)
							−0.477829	+	0.878453i	\(0.658576\pi\)
\(47\)	840656.	−	840656.i	0.172277	−	0.172277i	−0.615702	−	0.787979i	\(-0.711128\pi\)
							0.787979	+	0.615702i	\(0.211128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.9.c.a.12.8	✓	38
29.17	odd	4	inner	29.9.c.a.17.8	yes	38

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.9.c.a.12.8	✓	38	1.1	even	1	trivial
29.9.c.a.17.8	yes	38	29.17	odd	4	inner