Embedded newform 29.3.f.a.10.2

• Label: 29.3.f.a.10.2
• Level: $29$
• Weight: $3$
• Character: 29.10
• Analytic conductor: $0.790$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790192766645\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	29.10
Dual form			29.3.f.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29187 + 0.811733i) q^{2} +(2.15095 + 0.752648i) q^{3} +(-0.725528 + 1.50658i) q^{4} +(3.36294 + 0.767569i) q^{5} +(-3.38968 + 0.773673i) q^{6} +(-0.255710 + 0.123143i) q^{7} +(-0.968958 - 8.59974i) q^{8} +(-2.97640 - 2.37360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48q - 16q^{2} - 12q^{3} - 14q^{4} - 14q^{5} - 14q^{6} - 10q^{7} + 28q^{8} - 14q^{9} + O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29187	+	0.811733i	−0.645933	+	0.405867i	−0.814776	−	0.579776i	\(-0.803140\pi\)
							0.168843	+	0.985643i	\(0.445997\pi\)
\(3\)	2.15095	+	0.752648i	0.716982	+	0.250883i	0.664021	−	0.747714i	\(-0.268849\pi\)
							0.0529606	+	0.998597i	\(0.483134\pi\)
\(5\)	3.36294	+	0.767569i	0.672588	+	0.153514i	0.545158	−	0.838334i	\(-0.316470\pi\)
							0.127431	+	0.991847i	\(0.459327\pi\)
\(7\)	−0.255710	+	0.123143i	−0.0365299	+	0.0175919i	−0.452060	−	0.891988i	\(-0.649311\pi\)
							0.415530	+	0.909580i	\(0.363596\pi\)
\(11\)	1.62739	−	14.4435i	0.147945	−	1.31305i	−0.671978	−	0.740571i	\(-0.734555\pi\)
							0.819923	−	0.572474i	\(-0.194016\pi\)
\(13\)	−4.30976	+	3.43692i	−0.331520	+	0.264378i	−0.775076	−	0.631869i	\(-0.782288\pi\)
							0.443556	+	0.896247i	\(0.353717\pi\)
\(17\)	5.25353	+	5.25353i	0.309031	+	0.309031i	0.844534	−	0.535503i	\(-0.179878\pi\)
							−0.535503	+	0.844534i	\(0.679878\pi\)
\(19\)	9.56906	+	27.3468i	0.503635	+	1.43930i	0.862568	+	0.505942i	\(0.168855\pi\)
							−0.358933	+	0.933363i	\(0.616859\pi\)
\(23\)	−6.05686	−	26.5368i	−0.263342	−	1.15378i	−0.917600	−	0.397505i	\(-0.869876\pi\)
							0.654258	−	0.756271i	\(-0.272981\pi\)
\(29\)	2.32625	+	28.9065i	0.0802155	+	0.996778i
							\(31\)	−12.9513	+	8.13783i	−0.417783	+	0.262511i	−0.724479	−	0.689297i	\(-0.757920\pi\)
0.306695	+	0.951808i	\(0.400777\pi\)
\(37\)	3.05571	+	27.1202i	0.0825867	+	0.732977i	0.964955	+	0.262417i	\(0.0845196\pi\)
							−0.882368	+	0.470560i	\(0.844052\pi\)
\(41\)	−45.8791	+	45.8791i	−1.11900	+	1.11900i	−0.127114	+	0.991888i	\(0.540571\pi\)
							−0.991888	+	0.127114i	\(0.959429\pi\)
\(43\)	35.9543	−	57.2209i	0.836146	−	1.33072i	−0.105605	−	0.994408i	\(-0.533678\pi\)
							0.941751	−	0.336311i	\(-0.109179\pi\)
\(47\)	17.2732	+	1.94622i	0.367514	+	0.0414089i	0.293791	−	0.955870i	\(-0.405083\pi\)
							0.0737238	+	0.997279i	\(0.476512\pi\)