Embedded newform 74.6.f.a.9.5

• Label: 74.6.f.a.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	74.9
Dual form			74.6.f.a.33.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 + 3.93923i) q^{2} +(0.677795 - 3.84397i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(-48.1268 - 40.3832i) q^{5} +15.6131 q^{6} +(36.5328 + 30.6547i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(214.029 + 77.9001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{3} - 72 q^{5} + 114 q^{7} - 1536 q^{8} - 93 q^{9}+O(q^{10}) \)

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
							\(3\)	0.677795	−	3.84397i	0.0434806	−	0.246590i	−0.955319	−	0.295577i	\(-0.904488\pi\)
0.998799	+	0.0489866i	\(0.0155992\pi\)
\(5\)	−48.1268	−	40.3832i	−0.860919	−	0.722397i	0.101247	−	0.994861i	\(-0.467717\pi\)
							−0.962166	+	0.272465i	\(0.912161\pi\)
\(7\)	36.5328	+	30.6547i	0.281798	+	0.236457i	0.772720	−	0.634747i	\(-0.218896\pi\)
							−0.490922	+	0.871203i	\(0.663340\pi\)
\(11\)	208.255	+	360.709i	0.518937	+	0.898825i	0.999758	+	0.0220061i	\(0.00700533\pi\)
							−0.480821	+	0.876819i	\(0.659661\pi\)
\(13\)	830.062	−	302.118i	1.36224	−	0.495813i	0.445490	−	0.895287i	\(-0.353029\pi\)
							0.916745	+	0.399474i	\(0.130807\pi\)
\(17\)	819.360	+	298.223i	0.687627	+	0.250276i	0.662119	−	0.749399i	\(-0.269657\pi\)
							0.0255079	+	0.999675i	\(0.491880\pi\)
\(19\)	95.0587	−	539.105i	0.0604099	−	0.342601i	−0.939590	−	0.342302i	\(-0.888794\pi\)
							1.00000		0.000299662i	\(-9.53853e-5\pi\)
\(23\)	−64.9113	+	112.430i	−0.0255859	+	0.0443161i	−0.878535	−	0.477678i	\(-0.841478\pi\)
							0.852949	+	0.521994i	\(0.174812\pi\)
\(29\)	2256.30	+	3908.03i	0.498198	+	0.862905i	0.999998	−	0.00207919i	\(-0.000661826\pi\)
							−0.501800	+	0.864984i	\(0.667328\pi\)
\(31\)	3675.61			0.686951			0.343475	−	0.939162i	\(-0.388396\pi\)
							0.343475	+	0.939162i	\(0.388396\pi\)
\(37\)	−3529.67	+	7542.24i	−0.423868	+	0.905724i
							\(41\)	4234.33	−	1541.17i	0.393391	−	0.143183i	−0.137748	−	0.990467i	\(-0.543986\pi\)
0.531139	+	0.847285i	\(0.321764\pi\)
\(43\)	−6702.37			−0.552787			−0.276393	−	0.961045i	\(-0.589139\pi\)
							−0.276393	+	0.961045i	\(0.589139\pi\)
\(47\)	7935.28	−	13744.3i	0.523984	−	0.907567i	−0.475626	−	0.879647i	\(-0.657779\pi\)
							0.999610	−	0.0279192i	\(-0.00888810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.f.a.9.5	✓	48
37.33	even	9	inner	74.6.f.a.33.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.f.a.9.5	✓	48	1.1	even	1	trivial
74.6.f.a.33.5	yes	48	37.33	even	9	inner