Embedded newform 29.11.c.a.17.20

• Label: 29.11.c.a.17.20
• Level: $29$
• Weight: $11$
• Character: 29.17
• Analytic conductor: $18.425$
• Analytic rank: $0$
• Dimension: $48$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.4253603275\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.20
Character	\(\chi\)	\(=\)	29.17
Dual form			29.11.c.a.12.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(29.1517 - 29.1517i) q^{2} +(-96.9833 + 96.9833i) q^{3} -675.640i q^{4} -1089.69i q^{5} +5654.45i q^{6} -17667.0 q^{7} +(10155.3 + 10155.3i) q^{8} +40237.5i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{2} - 2 q^{3} - 4 q^{7} + 76770 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	29.1517	−	29.1517i	0.910990	−	0.910990i	−0.0853603	−	0.996350i	\(-0.527204\pi\)
							0.996350	+	0.0853603i	\(0.0272041\pi\)
\(3\)	−96.9833	+	96.9833i	−0.399108	+	0.399108i	−0.877918	−	0.478810i	\(-0.841068\pi\)
							0.478810	+	0.877918i	\(0.341068\pi\)
\(5\)		−	1089.69i		−	0.348700i	−0.984684	−	0.174350i	\(-0.944218\pi\)
							0.984684	−	0.174350i	\(-0.0557825\pi\)
\(7\)	−17667.0			−1.05117			−0.525583	−	0.850742i	\(-0.676153\pi\)
							−0.525583	+	0.850742i	\(0.676153\pi\)
\(11\)	66116.5	−	66116.5i	0.410532	−	0.410532i	−0.471392	−	0.881924i	\(-0.656248\pi\)
							0.881924	+	0.471392i	\(0.156248\pi\)
\(13\)			440355.i			1.18600i	0.805201	+	0.593002i	\(0.202057\pi\)
							−0.805201	+	0.593002i	\(0.797943\pi\)
\(17\)	−1.42241e6	+	1.42241e6i	−1.00180	+	1.00180i	−0.00180030	+	0.999998i	\(0.500573\pi\)
							−0.999998	+	0.00180030i	\(0.999427\pi\)
\(19\)	−3.09589e6	+	3.09589e6i	−1.25031	+	1.25031i	−0.294730	+	0.955581i	\(0.595230\pi\)
							−0.955581	+	0.294730i	\(0.904770\pi\)
\(23\)	1.03262e7			1.60436			0.802178	−	0.597085i	\(-0.203675\pi\)
							0.802178	+	0.597085i	\(0.203675\pi\)
\(29\)	5.16022e6	−	1.98514e7i	0.251581	−	0.967836i
							\(31\)	−2.89267e7	+	2.89267e7i	−1.01039	+	1.01039i	−0.0104476	+	0.999945i	\(0.503326\pi\)
−0.999945	+	0.0104476i	\(0.996674\pi\)
\(37\)	−2.36901e7	−	2.36901e7i	−0.341632	−	0.341632i	0.515348	−	0.856981i	\(-0.327663\pi\)
							−0.856981	+	0.515348i	\(0.827663\pi\)
\(41\)	−4.75736e6	−	4.75736e6i	−0.0410626	−	0.0410626i	0.686277	−	0.727340i	\(-0.259244\pi\)
							−0.727340	+	0.686277i	\(0.759244\pi\)
\(43\)	−1.29919e8	+	1.29919e8i	−0.883754	+	0.883754i	−0.993914	−	0.110160i	\(-0.964864\pi\)
							0.110160	+	0.993914i	\(0.464864\pi\)
\(47\)	−2.61280e8	−	2.61280e8i	−1.13924	−	1.13924i	−0.988586	−	0.150657i	\(-0.951861\pi\)
							−0.150657	−	0.988586i	\(-0.548139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.11.c.a.17.20	yes	48
29.12	odd	4	inner	29.11.c.a.12.20	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.11.c.a.12.20	✓	48	29.12	odd	4	inner
29.11.c.a.17.20	yes	48	1.1	even	1	trivial