Embedded newform 29.3.f.a.10.4

• Label: 29.3.f.a.10.4
• 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			10.4
Character	\(\chi\)	\(=\)	29.10
Dual form			29.3.f.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.05804 - 1.29315i) q^{2} +(-2.93183 - 1.02589i) q^{3} +(0.827740 - 1.71882i) q^{4} +(5.87720 + 1.34143i) q^{5} +(-7.36043 + 1.67997i) q^{6} +(-9.36468 + 4.50979i) q^{7} +(0.569384 + 5.05343i) q^{8} +(0.506667 + 0.404053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.05804	−	1.29315i	1.02902	−	0.646575i	0.0919649	−	0.995762i	\(-0.470685\pi\)
							0.937053	+	0.349188i	\(0.113542\pi\)
\(3\)	−2.93183	−	1.02589i	−0.977275	−	0.341963i	−0.206061	−	0.978539i	\(-0.566065\pi\)
							−0.771214	+	0.636576i	\(0.780350\pi\)
\(5\)	5.87720	+	1.34143i	1.17544	+	0.268286i	0.765262	−	0.643719i	\(-0.222610\pi\)
							0.410177	+	0.912006i	\(0.365467\pi\)
\(7\)	−9.36468	+	4.50979i	−1.33781	+	0.644256i	−0.959575	−	0.281452i	\(-0.909184\pi\)
							−0.378237	+	0.925709i	\(0.623470\pi\)
\(11\)	0.977326	−	8.67401i	0.0888478	−	0.788546i	−0.867723	−	0.497048i	\(-0.834417\pi\)
							0.956571	−	0.291499i	\(-0.0941540\pi\)
\(13\)	8.98997	−	7.16926i	0.691536	−	0.551482i	−0.213433	−	0.976958i	\(-0.568465\pi\)
							0.904970	+	0.425476i	\(0.139893\pi\)
\(17\)	−9.77422	−	9.77422i	−0.574954	−	0.574954i	0.358554	−	0.933509i	\(-0.383270\pi\)
							−0.933509	+	0.358554i	\(0.883270\pi\)
\(19\)	−4.49203	−	12.8375i	−0.236423	−	0.675658i	−0.999567	−	0.0294292i	\(-0.990631\pi\)
							0.763144	−	0.646228i	\(-0.223655\pi\)
\(23\)	6.44949	+	28.2571i	0.280413	+	1.22857i	0.897266	+	0.441490i	\(0.145550\pi\)
							−0.616853	+	0.787078i	\(0.711593\pi\)
\(29\)	−28.9618	+	1.48885i	−0.998681	+	0.0513395i
							\(31\)	32.8240	−	20.6247i	1.05884	−	0.665313i	0.114212	−	0.993456i	\(-0.463566\pi\)
0.944628	+	0.328144i	\(0.106423\pi\)
\(37\)	6.61609	+	58.7194i	0.178813	+	1.58701i	0.687581	+	0.726107i	\(0.258672\pi\)
							−0.508768	+	0.860904i	\(0.669899\pi\)
\(41\)	28.8990	−	28.8990i	0.704854	−	0.704854i	−0.260594	−	0.965448i	\(-0.583919\pi\)
							0.965448	+	0.260594i	\(0.0839185\pi\)
\(43\)	23.3701	−	37.1932i	0.543490	−	0.864959i	−0.456200	−	0.889877i	\(-0.650790\pi\)
							0.999689	+	0.0249182i	\(0.00793254\pi\)
\(47\)	−16.5298	−	1.86246i	−0.351698	−	0.0396269i	−0.0656518	−	0.997843i	\(-0.520913\pi\)
							−0.286047	+	0.958216i	\(0.592341\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.10.4	yes	48
3.2	odd	2		261.3.s.a.10.1		48
29.3	odd	28	inner	29.3.f.a.3.4	✓	48
87.32	even	28		261.3.s.a.235.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.3.4	✓	48	29.3	odd	28	inner
29.3.f.a.10.4	yes	48	1.1	even	1	trivial
261.3.s.a.10.1		48	3.2	odd	2
261.3.s.a.235.1		48	87.32	even	28