Embedded newform 29.10.e.a.4.2

• Label: 29.10.e.a.4.2
• 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			4.2
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-39.6416 - 9.04794i) q^{2} +(102.716 + 213.292i) q^{3} +(1028.30 + 495.202i) q^{4} +(153.222 - 671.311i) q^{5} +(-2141.98 - 9384.62i) q^{6} +(7874.00 - 3791.92i) q^{7} +(-20006.3 - 15954.5i) q^{8} +(-22670.8 + 28428.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−39.6416	−	9.04794i	−1.75193	−	0.399866i	−0.778280	−	0.627918i	\(-0.783907\pi\)
							−0.973649	+	0.228051i	\(0.926765\pi\)
\(3\)	102.716	+	213.292i	0.732138	+	1.52030i	0.849718	+	0.527237i	\(0.176772\pi\)
							−0.117580	+	0.993063i	\(0.537514\pi\)
\(5\)	153.222	−	671.311i	0.109637	−	0.480351i	−0.890062	−	0.455839i	\(-0.849339\pi\)
							0.999700	−	0.0245127i	\(-0.00780341\pi\)
\(7\)	7874.00	−	3791.92i	1.23952	−	0.596922i	0.304839	−	0.952404i	\(-0.401397\pi\)
							0.934683	+	0.355482i	\(0.115683\pi\)
\(11\)	38482.5	−	30688.7i	0.792494	−	0.631993i	−0.141234	−	0.989976i	\(-0.545107\pi\)
							0.933728	+	0.357984i	\(0.116536\pi\)
\(13\)	−78892.2	−	98927.7i	−0.766106	−	0.960666i	0.233827	−	0.972278i	\(-0.424875\pi\)
							−0.999933	+	0.0116118i	\(0.996304\pi\)
\(17\)		−	550356.i		−	1.59817i	−0.601217	−	0.799086i	\(-0.705317\pi\)
							0.601217	−	0.799086i	\(-0.294683\pi\)
\(19\)	15119.2	−	31395.4i	0.0266157	−	0.0552681i	−0.887237	−	0.461314i	\(-0.847378\pi\)
							0.913853	+	0.406046i	\(0.133093\pi\)
\(23\)	−327284.	−	1.43393e6i	−0.243865	−	1.06844i	−0.937464	−	0.348082i	\(-0.886833\pi\)
							0.693599	−	0.720361i	\(-0.256024\pi\)
\(29\)	−555175.	−	3.76815e6i	−0.145760	−	0.989320i
							\(31\)	−382619.	−	87330.2i	−0.0744113	−	0.0169839i	0.185153	−	0.982710i	\(-0.440722\pi\)
−0.259564	+	0.965726i	\(0.583579\pi\)
\(37\)	1.60992e7	+	1.28387e7i	1.41221	+	1.12620i	0.973797	+	0.227420i	\(0.0730291\pi\)
							0.438409	+	0.898776i	\(0.355542\pi\)
\(41\)		−	9.64239e6i		−	0.532914i	−0.963847	−	0.266457i	\(-0.914147\pi\)
							0.963847	−	0.266457i	\(-0.0858531\pi\)
\(43\)	−1.29560e7	+	2.95712e6i	−0.577914	+	0.131905i	−0.501479	−	0.865170i	\(-0.667211\pi\)
							−0.0764344	+	0.997075i	\(0.524354\pi\)
\(47\)	1.77899e7	−	1.41870e7i	0.531782	−	0.424082i	−0.320435	−	0.947270i	\(-0.603829\pi\)
							0.852217	+	0.523189i	\(0.175258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.2	✓	132
29.22	even	14	inner	29.10.e.a.22.2	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.2	✓	132	1.1	even	1	trivial
29.10.e.a.22.2	yes	132	29.22	even	14	inner