Embedded newform 74.6.f.a.9.1

• Label: 74.6.f.a.9.1
• Level: $74$
• Weight: $6$
• Character: 74.9
• Analytic conductor: $11.868$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8684026662\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.1
Character	\(\chi\)	\(=\)	74.9
Dual form			74.6.f.a.33.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 + 3.93923i) q^{2} +(-4.94687 + 28.0551i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(9.88188 + 8.29188i) q^{5} -113.952 q^{6} +(-125.671 - 105.450i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(-534.272 - 194.459i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{3} - 72 q^{5} + 114 q^{7} - 1536 q^{8} - 93 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.694593	+	3.93923i	0.122788	+	0.696364i
							\(3\)	−4.94687	+	28.0551i	−0.317342	+	1.79974i	0.241434	+	0.970417i	\(0.422382\pi\)
−0.558776	+	0.829319i	\(0.688729\pi\)
\(5\)	9.88188	+	8.29188i	0.176772	+	0.148330i	0.726881	−	0.686764i	\(-0.240969\pi\)
							−0.550109	+	0.835093i	\(0.685414\pi\)
\(7\)	−125.671	−	105.450i	−0.969370	−	0.813398i	0.0130822	−	0.999914i	\(-0.495836\pi\)
							−0.982452	+	0.186517i	\(0.940280\pi\)
\(11\)	343.698	+	595.303i	0.856437	+	1.48339i	0.875305	+	0.483571i	\(0.160660\pi\)
							−0.0188678	+	0.999822i	\(0.506006\pi\)
\(13\)	224.580	−	81.7403i	0.368563	−	0.134146i	−0.151097	−	0.988519i	\(-0.548281\pi\)
							0.519660	+	0.854373i	\(0.326058\pi\)
\(17\)	−144.077	−	52.4397i	−0.120913	−	0.0440087i	0.280855	−	0.959750i	\(-0.409382\pi\)
							−0.401768	+	0.915742i	\(0.631604\pi\)
\(19\)	196.190	−	1112.65i	0.124679	−	0.707090i	−0.856819	−	0.515617i	\(-0.827563\pi\)
							0.981498	−	0.191472i	\(-0.0613262\pi\)
\(23\)	−2391.34	+	4141.92i	−0.942587	+	1.63261i	−0.182075	+	0.983285i	\(0.558282\pi\)
							−0.760512	+	0.649324i	\(0.775052\pi\)
\(29\)	1131.94	+	1960.59i	0.249937	+	0.432903i	0.963508	−	0.267680i	\(-0.0862568\pi\)
							−0.713571	+	0.700583i	\(0.752924\pi\)
\(31\)	−5951.72			−1.11234			−0.556171	−	0.831068i	\(-0.687730\pi\)
							−0.556171	+	0.831068i	\(0.687730\pi\)
\(37\)	−3387.22	−	7607.28i	−0.406761	−	0.913534i
							\(41\)	−6901.43	+	2511.92i	−0.641179	+	0.233370i	−0.642090	−	0.766629i	\(-0.721932\pi\)
0.000910443		1.00000i	\(0.499710\pi\)
\(43\)	12501.2			1.03105			0.515525	−	0.856874i	\(-0.327597\pi\)
							0.515525	+	0.856874i	\(0.327597\pi\)
\(47\)	6032.09	−	10447.9i	0.398312	−	0.689896i	−0.595206	−	0.803573i	\(-0.702930\pi\)
							0.993518	+	0.113677i	\(0.0362630\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.f.a.9.1	✓	48
37.33	even	9	inner	74.6.f.a.33.1	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.f.a.9.1	✓	48	1.1	even	1	trivial
74.6.f.a.33.1	yes	48	37.33	even	9	inner