Embedded newform 29.3.f.a.11.4

• Label: 29.3.f.a.11.4
• Level: $29$
• Weight: $3$
• Character: 29.11
• 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			11.4
Character	\(\chi\)	\(=\)	29.11
Dual form			29.3.f.a.8.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65381 - 0.578694i) q^{2} +(-0.186386 - 1.65422i) q^{3} +(-0.727117 + 0.579856i) q^{4} +(-0.825315 + 1.71379i) q^{5} +(-1.26553 - 2.62791i) q^{6} +(-1.24782 + 1.56472i) q^{7} +(-4.59573 + 7.31406i) q^{8} +(6.07265 - 1.38604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{25}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65381	−	0.578694i	0.826906	−	0.289347i	0.116546	−	0.993185i	\(-0.462818\pi\)
							0.710360	+	0.703838i	\(0.248532\pi\)
\(3\)	−0.186386	−	1.65422i	−0.0621286	−	0.551406i	−0.985764	−	0.168134i	\(-0.946226\pi\)
							0.923635	−	0.383272i	\(-0.125203\pi\)
\(5\)	−0.825315	+	1.71379i	−0.165063	+	0.342757i	−0.967051	−	0.254583i	\(-0.918062\pi\)
							0.801988	+	0.597341i	\(0.203776\pi\)
\(7\)	−1.24782	+	1.56472i	−0.178260	+	0.223531i	−0.862932	−	0.505320i	\(-0.831374\pi\)
							0.684671	+	0.728852i	\(0.259946\pi\)
\(11\)	−6.63034	−	10.5521i	−0.602758	−	0.959284i	−0.999151	−	0.0412032i	\(-0.986881\pi\)
							0.396392	−	0.918081i	\(-0.370262\pi\)
\(13\)	−3.80625	−	0.868753i	−0.292789	−	0.0668271i	0.0736036	−	0.997288i	\(-0.476550\pi\)
							−0.366392	+	0.930460i	\(0.619407\pi\)
\(17\)	7.59392	−	7.59392i	0.446701	−	0.446701i	−0.447555	−	0.894256i	\(-0.647705\pi\)
							0.894256	+	0.447555i	\(0.147705\pi\)
\(19\)	−0.137534	−	0.0154964i	−0.00723865	−	0.000815600i	0.108344	−	0.994113i	\(-0.465445\pi\)
							−0.115583	+	0.993298i	\(0.536874\pi\)
\(23\)	26.7367	−	12.8757i	1.16247	−	0.559814i	0.249710	−	0.968321i	\(-0.419665\pi\)
							0.912755	+	0.408507i	\(0.133950\pi\)
\(29\)	8.15475	+	27.8298i	0.281198	+	0.959650i
							\(31\)	−54.1874	+	18.9610i	−1.74798	+	0.611645i	−0.998718	−	0.0506263i	\(-0.983878\pi\)
−0.749264	+	0.662272i	\(0.769593\pi\)
\(37\)	−29.9166	+	47.6121i	−0.808558	+	1.28681i	0.146149	+	0.989263i	\(0.453312\pi\)
							−0.954706	+	0.297550i	\(0.903831\pi\)
\(41\)	−25.9162	−	25.9162i	−0.632101	−	0.632101i	0.316493	−	0.948595i	\(-0.397494\pi\)
							−0.948595	+	0.316493i	\(0.897494\pi\)
\(43\)	5.16003	−	14.7465i	0.120001	−	0.342943i	−0.868177	−	0.496255i	\(-0.834708\pi\)
							0.988178	+	0.153312i	\(0.0489940\pi\)
\(47\)	55.9924	−	35.1824i	1.19133	−	0.748561i	0.217547	−	0.976050i	\(-0.430194\pi\)
							0.973781	+	0.227489i	\(0.0730516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.11.4	yes	48
3.2	odd	2		261.3.s.a.127.1		48
29.8	odd	28	inner	29.3.f.a.8.4	✓	48
87.8	even	28		261.3.s.a.37.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.8.4	✓	48	29.8	odd	28	inner
29.3.f.a.11.4	yes	48	1.1	even	1	trivial
261.3.s.a.37.1		48	87.8	even	28
261.3.s.a.127.1		48	3.2	odd	2