Embedded newform 29.10.e.a.4.13

• Label: 29.10.e.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.86626 + 2.02367i) q^{2} +(45.0762 + 93.6016i) q^{3} +(-386.781 - 186.264i) q^{4} +(69.9428 - 306.440i) q^{5} +(210.239 + 921.116i) q^{6} +(1930.08 - 929.476i) q^{7} +(-6692.78 - 5337.31i) q^{8} +(5542.75 - 6950.39i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.86626	+	2.02367i	0.391837	+	0.0894343i	0.413900	−	0.910322i	\(-0.364166\pi\)
							−0.0220629	+	0.999757i	\(0.507023\pi\)
\(3\)	45.0762	+	93.6016i	0.321293	+	0.667172i	0.997585	−	0.0694623i	\(-0.0221284\pi\)
							−0.676292	+	0.736634i	\(0.736414\pi\)
\(5\)	69.9428	−	306.440i	0.0500470	−	0.219270i	−0.943720	−	0.330745i	\(-0.892700\pi\)
							0.993767	+	0.111474i	\(0.0355573\pi\)
\(7\)	1930.08	−	929.476i	0.303832	−	0.146318i	−0.275755	−	0.961228i	\(-0.588928\pi\)
							0.579587	+	0.814910i	\(0.303214\pi\)
\(11\)	49029.5	−	39099.7i	1.00969	−	0.805205i	0.0287685	−	0.999586i	\(-0.490841\pi\)
							0.980926	+	0.194382i	\(0.0622700\pi\)
\(13\)	32910.8	+	41268.8i	0.319590	+	0.400753i	0.915513	−	0.402288i	\(-0.131785\pi\)
							−0.595923	+	0.803042i	\(0.703214\pi\)
\(17\)			251361.i			0.729926i	0.931022	+	0.364963i	\(0.118918\pi\)
							−0.931022	+	0.364963i	\(0.881082\pi\)
\(19\)	414347.	−	860401.i	0.729413	−	1.51464i	−0.123362	−	0.992362i	\(-0.539368\pi\)
							0.852775	−	0.522279i	\(-0.174918\pi\)
\(23\)	−397311.	−	1.74073e6i	−0.296043	−	1.29705i	−0.875964	−	0.482376i	\(-0.839774\pi\)
							0.579921	−	0.814673i	\(-0.303083\pi\)
\(29\)	24664.1	−	3.80874e6i	0.00647552	−	0.999979i
							\(31\)	−4.40534e6	−	1.00549e6i	−0.856746	−	0.195547i	−0.228483	−	0.973548i	\(-0.573377\pi\)
−0.628263	+	0.778001i	\(0.716234\pi\)
\(37\)	−9.30995e6	−	7.42444e6i	−0.816657	−	0.651262i	0.123370	−	0.992361i	\(-0.460630\pi\)
							−0.940028	+	0.341098i	\(0.889201\pi\)
\(41\)			324523.i			0.0179357i	0.999960	+	0.00896784i	\(0.00285459\pi\)
							−0.999960	+	0.00896784i	\(0.997145\pi\)
\(43\)	6.76926e6	−	1.54504e6i	0.301949	−	0.0689178i	−0.0688624	−	0.997626i	\(-0.521937\pi\)
							0.370811	+	0.928708i	\(0.379080\pi\)
\(47\)	−3.80724e7	+	3.03617e7i	−1.13807	+	0.907581i	−0.996603	−	0.0823615i	\(-0.973754\pi\)
							−0.141468	+	0.989943i	\(0.545182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.13	✓	132
29.22	even	14	inner	29.10.e.a.22.13	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.13	✓	132	1.1	even	1	trivial
29.10.e.a.22.13	yes	132	29.22	even	14	inner