Embedded newform 74.4.f.a.7.4

• Label: 74.4.f.a.7.4
• Level: $74$
• Weight: $4$
• Character: 74.7
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36614134042\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	74.7
Dual form			74.4.f.a.53.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.87939 - 0.684040i) q^{2} +(6.85178 + 2.49384i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-0.811775 - 4.60380i) q^{5} +14.5830 q^{6} +(-0.602158 - 3.41501i) q^{7} +(4.00000 - 6.92820i) q^{8} +(20.0444 + 16.8192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.87939	−	0.684040i	0.664463	−	0.241845i
							\(3\)	6.85178	+	2.49384i	1.31863	+	0.479940i	0.903018	−	0.429604i	\(-0.141347\pi\)
0.415608	+	0.909544i	\(0.363569\pi\)
\(5\)	−0.811775	−	4.60380i	−0.0726074	−	0.411777i	−0.999349	−	0.0360775i	\(-0.988514\pi\)
							0.926742	−	0.375699i	\(-0.122597\pi\)
\(7\)	−0.602158	−	3.41501i	−0.0325135	−	0.184393i	0.964226	−	0.265082i	\(-0.0853991\pi\)
							−0.996739	+	0.0806889i	\(0.974288\pi\)
\(11\)	−13.1138	+	22.7138i	−0.359452	+	0.622589i	−0.987869	−	0.155287i	\(-0.950370\pi\)
							0.628417	+	0.777876i	\(0.283703\pi\)
\(13\)	−28.7810	+	24.1502i	−0.614033	+	0.515235i	−0.895921	−	0.444212i	\(-0.853484\pi\)
							0.281889	+	0.959447i	\(0.409039\pi\)
\(17\)	8.68796	+	7.29007i	0.123949	+	0.104006i	0.702655	−	0.711530i	\(-0.251998\pi\)
							−0.578706	+	0.815536i	\(0.696442\pi\)
\(19\)	−2.89384	−	1.05327i	−0.0349417	−	0.0127177i	0.324490	−	0.945889i	\(-0.394807\pi\)
							−0.359432	+	0.933171i	\(0.617030\pi\)
\(23\)	−59.2395	−	102.606i	−0.537056	−	0.930209i	−0.999061	−	0.0433310i	\(-0.986203\pi\)
							0.462005	−	0.886878i	\(-0.347130\pi\)
\(29\)	−105.921	+	183.460i	−0.678240	+	1.17475i	0.297270	+	0.954793i	\(0.403924\pi\)
							−0.975511	+	0.219953i	\(0.929410\pi\)
\(31\)	−114.509			−0.663430			−0.331715	−	0.943380i	\(-0.607627\pi\)
							−0.331715	+	0.943380i	\(0.607627\pi\)
\(37\)	−185.046	+	128.106i	−0.822198	+	0.569201i
							\(41\)	−0.174333	+	0.146282i	−0.000664053	+	0.000557207i	−0.643120	−	0.765766i	\(-0.722360\pi\)
0.642456	+	0.766323i	\(0.277916\pi\)
\(43\)	176.917			0.627431			0.313715	−	0.949517i	\(-0.398426\pi\)
							0.313715	+	0.949517i	\(0.398426\pi\)
\(47\)	7.57890	+	13.1270i	0.0235212	+	0.0407399i	0.877546	−	0.479492i	\(-0.159179\pi\)
							−0.854025	+	0.520232i	\(0.825846\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.7.4	✓	24
37.16	even	9	inner	74.4.f.a.53.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.7.4	✓	24	1.1	even	1	trivial
74.4.f.a.53.4	yes	24	37.16	even	9	inner