Embedded newform 29.3.f.a.3.2

• Label: 29.3.f.a.3.2
• Level: $29$
• Weight: $3$
• Character: 29.3
• 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			3.2
Character	\(\chi\)	\(=\)	29.3
Dual form			29.3.f.a.10.2

$q$-expansion

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

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{28}\right)\)

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.3.2	✓	48
3.2	odd	2		261.3.s.a.235.3		48
29.10	odd	28	inner	29.3.f.a.10.2	yes	48
87.68	even	28		261.3.s.a.10.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.3.2	✓	48	1.1	even	1	trivial
29.3.f.a.10.2	yes	48	29.10	odd	28	inner
261.3.s.a.10.3		48	87.68	even	28
261.3.s.a.235.3		48	3.2	odd	2