Embedded newform 74.4.f.a.9.3

• Label: 74.4.f.a.9.3
• Level: $74$
• Weight: $4$
• Character: 74.9
• 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			9.3
Character	\(\chi\)	\(=\)	74.9
Dual form			74.4.f.a.33.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347296 - 1.96962i) q^{2} +(-0.0459350 + 0.260510i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(12.6348 + 10.6018i) q^{5} +0.529058 q^{6} +(0.748056 + 0.627693i) q^{7} +(4.00000 + 6.92820i) q^{8} +(25.3059 + 9.21061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.347296	−	1.96962i	−0.122788	−	0.696364i
							\(3\)	−0.0459350	+	0.260510i	−0.00884019	+	0.0501352i	−0.988909	−	0.148524i	\(-0.952548\pi\)
0.980069	+	0.198660i	\(0.0636588\pi\)
\(5\)	12.6348	+	10.6018i	1.13009	+	0.948257i	0.999070	−	0.0431233i	\(-0.0137308\pi\)
							0.131019	+	0.991380i	\(0.458175\pi\)
\(7\)	0.748056	+	0.627693i	0.0403912	+	0.0338923i	0.662760	−	0.748832i	\(-0.269385\pi\)
							−0.622369	+	0.782724i	\(0.713829\pi\)
\(11\)	−9.26296	−	16.0439i	−0.253899	−	0.439766i	0.710697	−	0.703498i	\(-0.248380\pi\)
							−0.964596	+	0.263732i	\(0.915046\pi\)
\(13\)	47.8263	−	17.4073i	1.02036	−	0.371379i	0.222955	−	0.974829i	\(-0.428430\pi\)
							0.797401	+	0.603450i	\(0.206208\pi\)
\(17\)	5.74073	+	2.08946i	0.0819019	+	0.0298099i	0.382646	−	0.923895i	\(-0.375013\pi\)
							−0.300744	+	0.953705i	\(0.597235\pi\)
\(19\)	0.257205	−	1.45868i	0.00310562	−	0.0176129i	−0.983216	−	0.182448i	\(-0.941598\pi\)
							0.986321	+	0.164835i	\(0.0527091\pi\)
\(23\)	−42.1910	+	73.0770i	−0.382497	+	0.662505i	−0.991419	−	0.130726i	\(-0.958269\pi\)
							0.608921	+	0.793231i	\(0.291603\pi\)
\(29\)	6.91180	+	11.9716i	0.0442583	+	0.0766576i	0.887306	−	0.461181i	\(-0.152574\pi\)
							−0.843048	+	0.537839i	\(0.819241\pi\)
\(31\)	5.83912			0.0338302			0.0169151	−	0.999857i	\(-0.494615\pi\)
							0.0169151	+	0.999857i	\(0.494615\pi\)
\(37\)	−217.685	+	57.1506i	−0.967222	+	0.253932i
							\(41\)	−194.400	+	70.7557i	−0.740491	+	0.269517i	−0.684599	−	0.728920i	\(-0.740023\pi\)
−0.0558925	+	0.998437i	\(0.517800\pi\)
\(43\)	−524.941			−1.86169			−0.930845	−	0.365413i	\(-0.880928\pi\)
							−0.930845	+	0.365413i	\(0.880928\pi\)
\(47\)	222.352	−	385.125i	0.690072	−	1.19524i	−0.281742	−	0.959490i	\(-0.590912\pi\)
							0.971814	−	0.235749i	\(-0.0757544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.9.3	✓	24
37.33	even	9	inner	74.4.f.a.33.3	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.9.3	✓	24	1.1	even	1	trivial
74.4.f.a.33.3	yes	24	37.33	even	9	inner