Embedded newform 29.15.c.a.12.7

• Label: 29.15.c.a.12.7
• Level: $29$
• Weight: $15$
• Character: 29.12
• Analytic conductor: $36.055$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	29.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.0554007641\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			12.7
Character	\(\chi\)	\(=\)	29.12
Dual form			29.15.c.a.17.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-120.665 - 120.665i) q^{2} +(-1296.13 - 1296.13i) q^{3} +12736.1i q^{4} -36154.9i q^{5} +312796. i q^{6} -444023. q^{7} +(-440176. + 440176. i) q^{8} -1.42305e6i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 312 q^{2} - 2 q^{3} - 4 q^{7} - 689310 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−120.665	−	120.665i	−0.942695	−	0.942695i	0.0557496	−	0.998445i	\(-0.482245\pi\)
							−0.998445	+	0.0557496i	\(0.982245\pi\)
\(3\)	−1296.13	−	1296.13i	−0.592653	−	0.592653i	0.345694	−	0.938347i	\(-0.387644\pi\)
							−0.938347	+	0.345694i	\(0.887644\pi\)
\(5\)		−	36154.9i		−	0.462783i	−0.972861	−	0.231391i	\(-0.925672\pi\)
							0.972861	−	0.231391i	\(-0.0743278\pi\)
\(7\)	−444023.			−0.539162			−0.269581	−	0.962978i	\(-0.586885\pi\)
							−0.269581	+	0.962978i	\(0.586885\pi\)
\(11\)	2.18317e6	+	2.18317e6i	0.112031	+	0.112031i	0.760900	−	0.648869i	\(-0.224758\pi\)
							−0.648869	+	0.760900i	\(0.724758\pi\)
\(13\)			2.23601e7i			0.356345i	0.983999	+	0.178173i	\(0.0570185\pi\)
							−0.983999	+	0.178173i	\(0.942981\pi\)
\(17\)	1.10126e8	+	1.10126e8i	0.268377	+	0.268377i	0.828446	−	0.560069i	\(-0.189225\pi\)
							−0.560069	+	0.828446i	\(0.689225\pi\)
\(19\)	6.60388e8	+	6.60388e8i	0.738795	+	0.738795i	0.972345	−	0.233550i	\(-0.0750342\pi\)
							−0.233550	+	0.972345i	\(0.575034\pi\)
\(23\)	5.00064e9			1.46869			0.734346	−	0.678775i	\(-0.237489\pi\)
							0.734346	+	0.678775i	\(0.237489\pi\)
\(29\)	−7.17237e9	+	1.56881e10i	−0.415793	+	0.909459i
							\(31\)	−7.63194e9	−	7.63194e9i	−0.277398	−	0.277398i	0.554672	−	0.832069i	\(-0.312844\pi\)
−0.832069	+	0.554672i	\(0.812844\pi\)
\(37\)	−2.66982e10	+	2.66982e10i	−0.281235	+	0.281235i	−0.833602	−	0.552366i	\(-0.813725\pi\)
							0.552366	+	0.833602i	\(0.313725\pi\)
\(41\)	1.14454e11	−	1.14454e11i	0.587683	−	0.587683i	−0.349320	−	0.937003i	\(-0.613587\pi\)
							0.937003	+	0.349320i	\(0.113587\pi\)
\(43\)	−1.95153e11	−	1.95153e11i	−0.717955	−	0.717955i	0.250231	−	0.968186i	\(-0.419493\pi\)
							−0.968186	+	0.250231i	\(0.919493\pi\)
\(47\)	−1.40007e11	+	1.40007e11i	−0.276353	+	0.276353i	−0.831651	−	0.555298i	\(-0.812604\pi\)
							0.555298	+	0.831651i	\(0.312604\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.15.c.a.12.7	✓	68
29.17	odd	4	inner	29.15.c.a.17.7	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.15.c.a.12.7	✓	68	1.1	even	1	trivial
29.15.c.a.17.7	yes	68	29.17	odd	4	inner