Embedded newform 74.4.f.a.33.1

• Label: 74.4.f.a.33.1
• Level: $74$
• Weight: $4$
• Character: 74.33
• 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			33.1
Character	\(\chi\)	\(=\)	74.33
Dual form			74.4.f.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347296 + 1.96962i) q^{2} +(-1.43352 - 8.12991i) q^{3} +(-3.75877 - 1.36808i) q^{4} +(-2.21967 + 1.86252i) q^{5} +16.5106 q^{6} +(-21.7866 + 18.2811i) q^{7} +(4.00000 - 6.92820i) q^{8} +(-38.6687 + 14.0742i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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
							\(3\)	−1.43352	−	8.12991i	−0.275881	−	1.56460i	−0.736148	−	0.676820i	\(-0.763357\pi\)
0.460267	−	0.887781i	\(-0.347754\pi\)
\(5\)	−2.21967	+	1.86252i	−0.198533	+	0.166589i	−0.736635	−	0.676291i	\(-0.763586\pi\)
							0.538101	+	0.842880i	\(0.319142\pi\)
\(7\)	−21.7866	+	18.2811i	−1.17637	+	0.987088i	−0.176369	+	0.984324i	\(0.556435\pi\)
							−0.999996	+	0.00276366i	\(0.999120\pi\)
\(11\)	−14.7962	+	25.6277i	−0.405565	+	0.702459i	−0.994387	−	0.105804i	\(-0.966258\pi\)
							0.588822	+	0.808262i	\(0.299592\pi\)
\(13\)	−35.3427	−	12.8637i	−0.754023	−	0.274442i	−0.0637252	−	0.997967i	\(-0.520298\pi\)
							−0.690297	+	0.723526i	\(0.742520\pi\)
\(17\)	31.2096	−	11.3594i	0.445261	−	0.162062i	−0.109652	−	0.993970i	\(-0.534974\pi\)
							0.554913	+	0.831908i	\(0.312751\pi\)
\(19\)	−21.5046	−	121.959i	−0.259658	−	1.47259i	−0.783828	−	0.620978i	\(-0.786736\pi\)
							0.524170	−	0.851614i	\(-0.324376\pi\)
\(23\)	−78.7697	−	136.433i	−0.714114	−	1.23688i	−0.963300	−	0.268425i	\(-0.913497\pi\)
							0.249187	−	0.968455i	\(-0.419837\pi\)
\(29\)	−73.5919	+	127.465i	−0.471230	+	0.816195i	−0.999458	−	0.0329076i	\(-0.989523\pi\)
							0.528228	+	0.849103i	\(0.322857\pi\)
\(31\)	36.4653			0.211270			0.105635	−	0.994405i	\(-0.466313\pi\)
							0.105635	+	0.994405i	\(0.466313\pi\)
\(37\)	200.463	+	102.312i	0.890698	+	0.454595i
							\(41\)	−325.103	−	118.328i	−1.23835	−	0.450724i	−0.361903	−	0.932216i	\(-0.617873\pi\)
−0.876451	+	0.481492i	\(0.840095\pi\)
\(43\)	160.652			0.569751			0.284875	−	0.958565i	\(-0.408048\pi\)
							0.284875	+	0.958565i	\(0.408048\pi\)
\(47\)	−271.116	−	469.587i	−0.841412	−	1.45737i	−0.888701	−	0.458487i	\(-0.848392\pi\)
							0.0472896	−	0.998881i	\(-0.484942\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.33.1	yes	24
37.9	even	9	inner	74.4.f.a.9.1	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.9.1	✓	24	37.9	even	9	inner
74.4.f.a.33.1	yes	24	1.1	even	1	trivial