Embedded newform 74.4.f.a.9.4

• Label: 74.4.f.a.9.4
• 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			9.4
Character	\(\chi\)	\(=\)	74.9
Dual form			74.4.f.a.33.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347296 - 1.96962i) q^{2} +(1.09285 - 6.19786i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-4.09539 - 3.43644i) q^{5} -12.5869 q^{6} +(-9.53718 - 8.00265i) q^{7} +(4.00000 + 6.92820i) q^{8} +(-11.8474 - 4.31210i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

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
							\(3\)	1.09285	−	6.19786i	0.210319	−	1.19278i	−0.678528	−	0.734574i	\(-0.737382\pi\)
0.888847	−	0.458204i	\(-0.151507\pi\)
\(5\)	−4.09539	−	3.43644i	−0.366303	−	0.307365i	0.440994	−	0.897510i	\(-0.354626\pi\)
							−0.807297	+	0.590145i	\(0.799070\pi\)
\(7\)	−9.53718	−	8.00265i	−0.514960	−	0.432102i	0.347911	−	0.937528i	\(-0.386891\pi\)
							−0.862870	+	0.505425i	\(0.831336\pi\)
\(11\)	1.97582	+	3.42221i	0.0541573	+	0.0938033i	0.891833	−	0.452365i	\(-0.149419\pi\)
							−0.837676	+	0.546168i	\(0.816086\pi\)
\(13\)	−17.2883	+	6.29244i	−0.368840	+	0.134247i	−0.519789	−	0.854295i	\(-0.673989\pi\)
							0.150949	+	0.988542i	\(0.451767\pi\)
\(17\)	−38.2570	−	13.9244i	−0.545805	−	0.198657i	0.0543769	−	0.998520i	\(-0.482683\pi\)
							−0.600181	+	0.799864i	\(0.704905\pi\)
\(19\)	10.3916	−	58.9337i	0.125474	−	0.711596i	−0.855552	−	0.517717i	\(-0.826782\pi\)
							0.981025	−	0.193879i	\(-0.0621068\pi\)
\(23\)	33.3338	−	57.7358i	0.302199	−	0.523424i	−0.674435	−	0.738334i	\(-0.735613\pi\)
							0.976634	+	0.214910i	\(0.0689460\pi\)
\(29\)	7.55458	+	13.0849i	0.0483742	+	0.0837865i	0.889199	−	0.457521i	\(-0.151263\pi\)
							−0.840824	+	0.541308i	\(0.817929\pi\)
\(31\)	186.847			1.08254			0.541269	−	0.840850i	\(-0.317944\pi\)
							0.541269	+	0.840850i	\(0.317944\pi\)
\(37\)	200.465	−	102.306i	0.890711	−	0.454570i
							\(41\)	−115.413	+	42.0070i	−0.439623	+	0.160010i	−0.552343	−	0.833617i	\(-0.686266\pi\)
0.112720	+	0.993627i	\(0.464044\pi\)
\(43\)	257.678			0.913849			0.456925	−	0.889505i	\(-0.348951\pi\)
							0.456925	+	0.889505i	\(0.348951\pi\)
\(47\)	87.2301	−	151.087i	0.270720	−	0.468900i	−0.698327	−	0.715779i	\(-0.746072\pi\)
							0.969046	+	0.246879i	\(0.0794050\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.9.4	✓	24
37.33	even	9	inner	74.4.f.a.33.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.9.4	✓	24	1.1	even	1	trivial
74.4.f.a.33.4	yes	24	37.33	even	9	inner