Embedded newform 74.4.f.a.53.4

• Label: 74.4.f.a.53.4
• Level: $74$
• Weight: $4$
• Character: 74.53
• 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			53.4
Character	\(\chi\)	\(=\)	74.53
Dual form			74.4.f.a.7.4

$q$-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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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	0.415608	−	0.909544i	\(-0.363569\pi\)
0.903018	+	0.429604i	\(0.141347\pi\)
\(5\)	−0.811775	+	4.60380i	−0.0726074	+	0.411777i	0.926742	+	0.375699i	\(0.122597\pi\)
							−0.999349	+	0.0360775i	\(0.988514\pi\)
\(7\)	−0.602158	+	3.41501i	−0.0325135	+	0.184393i	−0.996739	−	0.0806889i	\(-0.974288\pi\)
							0.964226	+	0.265082i	\(0.0853991\pi\)
\(11\)	−13.1138	−	22.7138i	−0.359452	−	0.622589i	0.628417	−	0.777876i	\(-0.283703\pi\)
							−0.987869	+	0.155287i	\(0.950370\pi\)
\(13\)	−28.7810	−	24.1502i	−0.614033	−	0.515235i	0.281889	−	0.959447i	\(-0.409039\pi\)
							−0.895921	+	0.444212i	\(0.853484\pi\)
\(17\)	8.68796	−	7.29007i	0.123949	−	0.104006i	−0.578706	−	0.815536i	\(-0.696442\pi\)
							0.702655	+	0.711530i	\(0.251998\pi\)
\(19\)	−2.89384	+	1.05327i	−0.0349417	+	0.0127177i	−0.359432	−	0.933171i	\(-0.617030\pi\)
							0.324490	+	0.945889i	\(0.394807\pi\)
\(23\)	−59.2395	+	102.606i	−0.537056	+	0.930209i	0.462005	+	0.886878i	\(0.347130\pi\)
							−0.999061	+	0.0433310i	\(0.986203\pi\)
\(29\)	−105.921	−	183.460i	−0.678240	−	1.17475i	−0.975511	−	0.219953i	\(-0.929410\pi\)
							0.297270	−	0.954793i	\(-0.403924\pi\)
\(31\)	−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	0.642456	−	0.766323i	\(-0.277916\pi\)
−0.643120	+	0.765766i	\(0.722360\pi\)
\(43\)	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	−0.854025	−	0.520232i	\(-0.825846\pi\)
							0.877546	+	0.479492i	\(0.159179\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.7.4	✓	24	37.7	even	9	inner
74.4.f.a.53.4	yes	24	1.1	even	1	trivial