Embedded newform 29.10.e.a.4.9

• Label: 29.10.e.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	29.4
Dual form			29.10.e.a.22.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.0968 - 3.21750i) q^{2} +(63.5414 + 131.945i) q^{3} +(-272.929 - 131.436i) q^{4} +(30.5824 - 133.990i) q^{5} +(-471.197 - 2064.45i) q^{6} +(5267.91 - 2536.89i) q^{7} +(9212.56 + 7346.77i) q^{8} +(-1099.86 + 1379.18i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 7 q^{2} - 7 q^{3} + 5797 q^{4} + 1367 q^{5} - 7607 q^{6} + 4949 q^{7} + 31766 q^{8} + 112237 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.0968	−	3.21750i	−0.622996	−	0.142195i	−0.100641	−	0.994923i	\(-0.532089\pi\)
							−0.522355	+	0.852728i	\(0.674946\pi\)
\(3\)	63.5414	+	131.945i	0.452909	+	0.940476i	0.994973	+	0.100139i	\(0.0319287\pi\)
							−0.542064	+	0.840337i	\(0.682357\pi\)
\(5\)	30.5824	−	133.990i	0.0218830	−	0.0958757i	−0.962807	−	0.270190i	\(-0.912914\pi\)
							0.984690	+	0.174314i	\(0.0557707\pi\)
\(7\)	5267.91	−	2536.89i	0.829272	−	0.399356i	0.0294298	−	0.999567i	\(-0.490631\pi\)
							0.799842	+	0.600211i	\(0.204917\pi\)
\(11\)	−20061.9	+	15998.8i	−0.413147	+	0.329474i	−0.807906	−	0.589311i	\(-0.799399\pi\)
							0.394760	+	0.918784i	\(0.370828\pi\)
\(13\)	−13147.4	−	16486.3i	−0.127672	−	0.160095i	0.713887	−	0.700261i	\(-0.246933\pi\)
							−0.841559	+	0.540166i	\(0.818362\pi\)
\(17\)			339168.i			0.984905i	0.870339	+	0.492453i	\(0.163900\pi\)
							−0.870339	+	0.492453i	\(0.836100\pi\)
\(19\)	−236915.	+	491959.i	−0.417063	+	0.866040i	0.581556	+	0.813506i	\(0.302444\pi\)
							−0.998619	+	0.0525340i	\(0.983270\pi\)
\(23\)	419694.	+	1.83880e6i	0.312722	+	1.37012i	0.850029	+	0.526736i	\(0.176584\pi\)
							−0.537307	+	0.843387i	\(0.680558\pi\)
\(29\)	−3.39044e6	+	1.73554e6i	−0.890153	+	0.455662i
							\(31\)	−2.04027e6	−	465678.i	−0.396789	−	0.0905646i	0.0194702	−	0.999810i	\(-0.493802\pi\)
−0.416259	+	0.909246i	\(0.636659\pi\)
\(37\)	3.12417e6	+	2.49144e6i	0.274048	+	0.218546i	0.750864	−	0.660457i	\(-0.229638\pi\)
							−0.476815	+	0.879004i	\(0.658209\pi\)
\(41\)			1.26163e7i			0.697274i	0.937258	+	0.348637i	\(0.113355\pi\)
							−0.937258	+	0.348637i	\(0.886645\pi\)
\(43\)	8.34630e6	−	1.90499e6i	0.372294	−	0.0849737i	−0.0322802	−	0.999479i	\(-0.510277\pi\)
							0.404574	+	0.914505i	\(0.367420\pi\)
\(47\)	1.82345e7	−	1.45415e7i	0.545071	−	0.434680i	−0.311847	−	0.950132i	\(-0.600948\pi\)
							0.856918	+	0.515453i	\(0.172376\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.10.e.a.4.9	✓	132
29.22	even	14	inner	29.10.e.a.22.9	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.10.e.a.4.9	✓	132	1.1	even	1	trivial
29.10.e.a.22.9	yes	132	29.22	even	14	inner