Embedded newform 74.6.f.a.9.7

• Label: 74.6.f.a.9.7
• 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			9.7
Character	\(\chi\)	\(=\)	74.9
Dual form			74.6.f.a.33.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 + 3.93923i) q^{2} +(3.82058 - 21.6676i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(-48.1841 - 40.4312i) q^{5} +88.0075 q^{6} +(134.280 + 112.674i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(-226.543 - 82.4550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{3} - 72 q^{5} + 114 q^{7} - 1536 q^{8} - 93 q^{9}+O(q^{10}) \)

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
							\(3\)	3.82058	−	21.6676i	0.245091	−	1.38998i	−0.575191	−	0.818019i	\(-0.695072\pi\)
0.820282	−	0.571959i	\(-0.193816\pi\)
\(5\)	−48.1841	−	40.4312i	−0.861943	−	0.723256i	0.100443	−	0.994943i	\(-0.467974\pi\)
							−0.962386	+	0.271687i	\(0.912418\pi\)
\(7\)	134.280	+	112.674i	1.03578	+	0.869121i	0.991527	−	0.129899i	\(-0.0414654\pi\)
							0.0442511	+	0.999020i	\(0.485910\pi\)
\(11\)	−241.528	−	418.338i	−0.601846	−	1.04243i	−0.992541	−	0.121907i	\(-0.961099\pi\)
							0.390696	−	0.920520i	\(-0.372234\pi\)
\(13\)	−625.526	+	227.673i	−1.02657	+	0.373640i	−0.799773	−	0.600303i	\(-0.795047\pi\)
							−0.226794	+	0.973943i	\(0.572824\pi\)
\(17\)	−2168.57	−	789.295i	−1.81991	−	0.662395i	−0.995316	−	0.0966759i	\(-0.969179\pi\)
							−0.824598	−	0.565719i	\(-0.808599\pi\)
\(19\)	−145.652	+	826.036i	−0.0925622	+	0.524946i	0.902905	+	0.429840i	\(0.141430\pi\)
							−0.995467	+	0.0951060i	\(0.969681\pi\)
\(23\)	−2158.59	+	3738.78i	−0.850844	+	1.47370i	0.0296040	+	0.999562i	\(0.490575\pi\)
							−0.880448	+	0.474143i	\(0.842758\pi\)
\(29\)	−2189.79	−	3792.82i	−0.483511	−	0.837466i	0.516309	−	0.856402i	\(-0.327305\pi\)
							−0.999821	+	0.0189361i	\(0.993972\pi\)
\(31\)	5343.50			0.998670			0.499335	−	0.866409i	\(-0.333578\pi\)
							0.499335	+	0.866409i	\(0.333578\pi\)
\(37\)	5085.14	−	6594.34i	0.610659	−	0.791894i
							\(41\)	1969.85	−	716.968i	0.183010	−	0.0666101i	−0.248890	−	0.968532i	\(-0.580066\pi\)
0.431900	+	0.901922i	\(0.357843\pi\)
\(43\)	8766.22			0.723005			0.361502	−	0.932371i	\(-0.382264\pi\)
							0.361502	+	0.932371i	\(0.382264\pi\)
\(47\)	3512.58	−	6083.97i	0.231943	−	0.401737i	−0.726437	−	0.687233i	\(-0.758825\pi\)
							0.958380	+	0.285496i	\(0.0921583\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.6.f.a.9.7	✓	48
37.33	even	9	inner	74.6.f.a.33.7	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.6.f.a.9.7	✓	48	1.1	even	1	trivial
74.6.f.a.33.7	yes	48	37.33	even	9	inner