Embedded newform 29.13.c.a.12.16

• Label: 29.13.c.a.12.16
• Level: $29$
• Weight: $13$
• Character: 29.12
• Analytic conductor: $26.506$
• Analytic rank: $0$
• Dimension: $58$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.5058207010\)
Analytic rank:	\(0\)
Dimension:	\(58\)
Relative dimension:	\(29\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.16
Character	\(\chi\)	\(=\)	29.12
Dual form			29.13.c.a.17.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.76721 + 8.76721i) q^{2} +(327.395 + 327.395i) q^{3} -3942.27i q^{4} +12032.8i q^{5} +5740.68i q^{6} -63314.5 q^{7} +(70473.2 - 70473.2i) q^{8} -317066. i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 58 q + 88 q^{2} - 2 q^{3} - 4 q^{7} + 79650 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.76721	+	8.76721i	0.136988	+	0.136988i	0.772275	−	0.635288i	\(-0.219119\pi\)
							−0.635288	+	0.772275i	\(0.719119\pi\)
\(3\)	327.395	+	327.395i	0.449102	+	0.449102i	0.895056	−	0.445954i	\(-0.147136\pi\)
							−0.445954	+	0.895056i	\(0.647136\pi\)
\(5\)			12032.8i			0.770100i	0.922896	+	0.385050i	\(0.125816\pi\)
							−0.922896	+	0.385050i	\(0.874184\pi\)
\(7\)	−63314.5			−0.538164			−0.269082	−	0.963117i	\(-0.586720\pi\)
							−0.269082	+	0.963117i	\(0.586720\pi\)
\(11\)	481696.	+	481696.i	0.271905	+	0.271905i	0.829867	−	0.557962i	\(-0.188416\pi\)
							−0.557962	+	0.829867i	\(0.688416\pi\)
\(13\)		−	2.97275e6i		−	0.615883i	−0.951405	−	0.307942i	\(-0.900360\pi\)
							0.951405	−	0.307942i	\(-0.0996401\pi\)
\(17\)	−1.19355e7	−	1.19355e7i	−0.494480	−	0.494480i	0.415235	−	0.909714i	\(-0.363699\pi\)
							−0.909714	+	0.415235i	\(0.863699\pi\)
\(19\)	−9.65684e6	−	9.65684e6i	−0.205264	−	0.205264i	0.596987	−	0.802251i	\(-0.296364\pi\)
							−0.802251	+	0.596987i	\(0.796364\pi\)
\(23\)	1.80656e7			0.122035			0.0610176	−	0.998137i	\(-0.480565\pi\)
							0.0610176	+	0.998137i	\(0.480565\pi\)
\(29\)	9.12532e7	−	5.87782e8i	0.153412	−	0.988162i
							\(31\)	−1.25367e9	−	1.25367e9i	−1.41258	−	1.41258i	−0.740247	−	0.672335i	\(-0.765292\pi\)
−0.672335	−	0.740247i	\(-0.734708\pi\)
\(37\)	1.88341e9	−	1.88341e9i	0.734066	−	0.734066i	−0.237356	−	0.971423i	\(-0.576281\pi\)
							0.971423	+	0.237356i	\(0.0762809\pi\)
\(41\)	4.96087e9	−	4.96087e9i	1.04437	−	1.04437i	0.0454029	−	0.998969i	\(-0.485543\pi\)
							0.998969	−	0.0454029i	\(-0.0144572\pi\)
\(43\)	−2.11121e9	−	2.11121e9i	−0.333980	−	0.333980i	0.520116	−	0.854096i	\(-0.325889\pi\)
							−0.854096	+	0.520116i	\(0.825889\pi\)
\(47\)	−2.38035e9	+	2.38035e9i	−0.220828	+	0.220828i	−0.808847	−	0.588019i	\(-0.799908\pi\)
							0.588019	+	0.808847i	\(0.299908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.13.c.a.12.16	✓	58
29.17	odd	4	inner	29.13.c.a.17.16	yes	58

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.13.c.a.12.16	✓	58	1.1	even	1	trivial
29.13.c.a.17.16	yes	58	29.17	odd	4	inner