Embedded newform 29.3.f.a.10.1

• Label: 29.3.f.a.10.1
• 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			10.1
Character	\(\chi\)	\(=\)	29.10
Dual form			29.3.f.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.24379 + 2.03821i) q^{2} +(-2.23035 - 0.780434i) q^{3} +(4.63233 - 9.61914i) q^{4} +(-5.12808 - 1.17045i) q^{5} +(8.82547 - 2.01436i) q^{6} +(-6.56728 + 3.16264i) q^{7} +(2.86375 + 25.4165i) q^{8} +(-2.67109 - 2.13013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.24379	+	2.03821i	−1.62189	+	1.01910i	−0.657926	+	0.753082i	\(0.728566\pi\)
							−0.963967	+	0.266021i	\(0.914291\pi\)
\(3\)	−2.23035	−	0.780434i	−0.743450	−	0.260145i	−0.0681376	−	0.997676i	\(-0.521706\pi\)
							−0.675313	+	0.737531i	\(0.735991\pi\)
\(5\)	−5.12808	−	1.17045i	−1.02562	−	0.234090i	−0.323557	−	0.946209i	\(-0.604879\pi\)
							−0.702059	+	0.712119i	\(0.747736\pi\)
\(7\)	−6.56728	+	3.16264i	−0.938183	+	0.451805i	−0.839528	−	0.543317i	\(-0.817168\pi\)
							−0.0986550	+	0.995122i	\(0.531454\pi\)
\(11\)	−0.480789	+	4.26712i	−0.0437081	+	0.387920i	0.952745	+	0.303771i	\(0.0982458\pi\)
							−0.996453	+	0.0841491i	\(0.973183\pi\)
\(13\)	2.73044	−	2.17745i	0.210034	−	0.167496i	−0.512823	−	0.858494i	\(-0.671400\pi\)
							0.722856	+	0.690998i	\(0.242829\pi\)
\(17\)	−6.15599	−	6.15599i	−0.362117	−	0.362117i	0.502475	−	0.864592i	\(-0.332423\pi\)
							−0.864592	+	0.502475i	\(0.832423\pi\)
\(19\)	5.18330	+	14.8130i	0.272806	+	0.779633i	0.996030	+	0.0890201i	\(0.0283736\pi\)
							−0.723224	+	0.690613i	\(0.757341\pi\)
\(23\)	−3.58915	−	15.7251i	−0.156050	−	0.683699i	−0.991055	−	0.133457i	\(-0.957392\pi\)
							0.835005	−	0.550243i	\(-0.185465\pi\)
\(29\)	−0.661611	−	28.9925i	−0.0228142	−	0.999740i
							\(31\)	−34.6988	+	21.8027i	−1.11931	+	0.703312i	−0.958902	−	0.283738i	\(-0.908425\pi\)
−0.160413	+	0.987050i	\(0.551283\pi\)
\(37\)	−5.62751	−	49.9455i	−0.152095	−	1.34988i	−0.805376	−	0.592764i	\(-0.798037\pi\)
							0.653281	−	0.757115i	\(-0.273392\pi\)
\(41\)	−0.940907	+	0.940907i	−0.0229490	+	0.0229490i	−0.718488	−	0.695539i	\(-0.755166\pi\)
							0.695539	+	0.718488i	\(0.255166\pi\)
\(43\)	−10.2578	+	16.3252i	−0.238554	+	0.379657i	−0.944448	−	0.328661i	\(-0.893403\pi\)
							0.705894	+	0.708318i	\(0.250546\pi\)
\(47\)	−59.1758	−	6.66751i	−1.25906	−	0.141862i	−0.542899	−	0.839798i	\(-0.682673\pi\)
							−0.716160	+	0.697936i	\(0.754102\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.10.1	yes	48
3.2	odd	2		261.3.s.a.10.4		48
29.3	odd	28	inner	29.3.f.a.3.1	✓	48
87.32	even	28		261.3.s.a.235.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.3.1	✓	48	29.3	odd	28	inner
29.3.f.a.10.1	yes	48	1.1	even	1	trivial
261.3.s.a.10.4		48	3.2	odd	2
261.3.s.a.235.4		48	87.32	even	28