Embedded newform 29.10.e.a.4.14

• Label: 29.10.e.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.94046 + 2.04060i) q^{2} +(101.023 + 209.777i) q^{3} +(-385.528 - 185.661i) q^{4} +(456.096 - 1998.29i) q^{5} +(475.124 + 2081.65i) q^{6} +(-6625.33 + 3190.59i) q^{7} +(-6738.82 - 5374.03i) q^{8} +(-21528.6 + 26996.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.94046	+	2.04060i	0.395116	+	0.0901827i	0.415463	−	0.909610i	\(-0.363620\pi\)
							−0.0203463	+	0.999793i	\(0.506477\pi\)
\(3\)	101.023	+	209.777i	0.720073	+	1.49525i	0.862833	+	0.505489i	\(0.168688\pi\)
							−0.142761	+	0.989757i	\(0.545598\pi\)
\(5\)	456.096	−	1998.29i	0.326356	−	1.42986i	−0.499665	−	0.866219i	\(-0.666544\pi\)
							0.826021	−	0.563640i	\(-0.190599\pi\)
\(7\)	−6625.33	+	3190.59i	−1.04296	+	0.502262i	−0.875298	−	0.483583i	\(-0.839335\pi\)
							−0.167659	+	0.985845i	\(0.553621\pi\)
\(11\)	−39537.3	+	31529.9i	−0.814216	+	0.649315i	−0.939402	−	0.342818i	\(-0.888619\pi\)
							0.125186	+	0.992133i	\(0.460047\pi\)
\(13\)	−79878.2	−	100164.i	−0.775681	−	0.972673i	0.224317	−	0.974516i	\(-0.427985\pi\)
							−0.999998	+	0.00184274i	\(0.999413\pi\)
\(17\)		−	563559.i		−	1.63651i	−0.574854	−	0.818256i	\(-0.694941\pi\)
							0.574854	−	0.818256i	\(-0.305059\pi\)
\(19\)	−150381.	+	312269.i	−0.264729	+	0.549715i	−0.990384	−	0.138343i	\(-0.955822\pi\)
							0.725656	+	0.688058i	\(0.241537\pi\)
\(23\)	110196.	+	482801.i	0.0821092	+	0.359744i	0.999246	−	0.0388291i	\(-0.0123628\pi\)
							−0.917137	+	0.398573i	\(0.869506\pi\)
\(29\)	3.71882e6	+	823113.i	0.976370	+	0.216107i
							\(31\)	−4.39077e6	−	1.00216e6i	−0.853911	−	0.194900i	−0.226908	−	0.973916i	\(-0.572862\pi\)
−0.627003	+	0.779017i	\(0.715719\pi\)
\(37\)	−3.77806e6	−	3.01291e6i	−0.331407	−	0.264288i	0.443622	−	0.896214i	\(-0.353693\pi\)
							−0.775029	+	0.631926i	\(0.782265\pi\)
\(41\)			3.19128e7i			1.76375i	0.471484	+	0.881875i	\(0.343719\pi\)
							−0.471484	+	0.881875i	\(0.656281\pi\)
\(43\)	−3.40951e7	+	7.78198e6i	−1.52084	+	0.347122i	−0.899676	−	0.436558i	\(-0.856198\pi\)
							−0.621165	+	0.783680i	\(0.713340\pi\)
\(47\)	−4.93653e6	+	3.93675e6i	−0.147564	+	0.117679i	−0.694489	−	0.719503i	\(-0.744370\pi\)
							0.546925	+	0.837182i	\(0.315798\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.14	✓	132
29.22	even	14	inner	29.10.e.a.22.14	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.14	✓	132	1.1	even	1	trivial
29.10.e.a.22.14	yes	132	29.22	even	14	inner