Embedded newform 29.10.e.a.4.7

• Label: 29.10.e.a.4.7
• Level: $29$
• Weight: $10$
• Character: 29.4
• Analytic conductor: $14.936$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.9360392488\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			4.7
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.8398 - 4.75655i) q^{2} +(-98.0825 - 203.670i) q^{3} +(-49.6235 - 23.8974i) q^{4} +(-133.957 + 586.902i) q^{5} +(1075.25 + 4710.98i) q^{6} +(-5718.27 + 2753.77i) q^{7} +(9477.13 + 7557.76i) q^{8} +(-19589.3 + 24564.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.8398	−	4.75655i	−0.920998	−	0.210212i	−0.264365	−	0.964423i	\(-0.585163\pi\)
							−0.656632	+	0.754211i	\(0.728020\pi\)
\(3\)	−98.0825	−	203.670i	−0.699111	−	1.45172i	−0.883291	−	0.468826i	\(-0.844677\pi\)
							0.184180	−	0.982892i	\(-0.441037\pi\)
\(5\)	−133.957	+	586.902i	−0.0958516	+	0.419953i	−0.999973	−	0.00732205i	\(-0.997669\pi\)
							0.904122	+	0.427275i	\(0.140526\pi\)
\(7\)	−5718.27	+	2753.77i	−0.900168	+	0.433498i	−0.825950	−	0.563744i	\(-0.809360\pi\)
							−0.0742183	+	0.997242i	\(0.523646\pi\)
\(11\)	−11787.6	+	9400.28i	−0.242749	+	0.193586i	−0.737305	−	0.675560i	\(-0.763902\pi\)
							0.494556	+	0.869146i	\(0.335331\pi\)
\(13\)	−93238.2	−	116917.i	−0.905417	−	1.13536i	−0.990297	−	0.138966i	\(-0.955622\pi\)
							0.0848803	−	0.996391i	\(-0.472949\pi\)
\(17\)			374159.i			1.08652i	0.839565	+	0.543259i	\(0.182810\pi\)
							−0.839565	+	0.543259i	\(0.817190\pi\)
\(19\)	175333.	−	364083.i	0.308654	−	0.640928i	−0.687723	−	0.725974i	\(-0.741389\pi\)
							0.996377	+	0.0850461i	\(0.0271037\pi\)
\(23\)	−113212.	−	496012.i	−0.0843559	−	0.369587i	0.915076	−	0.403281i	\(-0.132130\pi\)
							−0.999432	+	0.0336934i	\(0.989273\pi\)
\(29\)	−669921.	−	3.74945e6i	−0.175887	−	0.984410i
							\(31\)	−4.78234e6	−	1.09154e6i	−0.930064	−	0.212281i	−0.269456	−	0.963013i	\(-0.586844\pi\)
−0.660607	+	0.750732i	\(0.729701\pi\)
\(37\)	783393.	+	624735.i	0.0687182	+	0.0548010i	0.657247	−	0.753676i	\(-0.271721\pi\)
							−0.588528	+	0.808477i	\(0.700292\pi\)
\(41\)			2.96208e7i			1.63708i	0.574450	+	0.818539i	\(0.305216\pi\)
							−0.574450	+	0.818539i	\(0.694784\pi\)
\(43\)	3.73565e6	−	852637.i	0.166632	−	0.0380326i	−0.138391	−	0.990378i	\(-0.544193\pi\)
							0.305023	+	0.952345i	\(0.401336\pi\)
\(47\)	3.73550e7	−	2.97896e7i	1.11663	−	0.890481i	0.121851	−	0.992548i	\(-0.461117\pi\)
							0.994778	+	0.102067i	\(0.0325456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.7	✓	132
29.22	even	14	inner	29.10.e.a.22.7	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.7	✓	132	1.1	even	1	trivial
29.10.e.a.22.7	yes	132	29.22	even	14	inner