Embedded newform 29.3.f.a.3.1

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

$q$-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 \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.3.1	✓	48	1.1	even	1	trivial
29.3.f.a.10.1	yes	48	29.10	odd	28	inner
261.3.s.a.10.4		48	87.68	even	28
261.3.s.a.235.4		48	3.2	odd	2