Embedded newform 74.4.f.a.49.3

• Label: 74.4.f.a.49.3
• Level: $74$
• Weight: $4$
• Character: 74.49
• 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			49.3
Character	\(\chi\)	\(=\)	74.49
Dual form			74.4.f.a.71.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53209 + 1.28558i) q^{2} +(3.31240 + 2.77944i) q^{3} +(0.694593 - 3.93923i) q^{4} +(10.2218 - 3.72042i) q^{5} -8.64807 q^{6} +(12.8394 - 4.67315i) q^{7} +(4.00000 + 6.92820i) q^{8} +(-1.44175 - 8.17656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 6 q^{3} - 9 q^{5} + 36 q^{6} - 75 q^{7} + 96 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.53209	+	1.28558i	−0.541675	+	0.454519i
							\(3\)	3.31240	+	2.77944i	0.637473	+	0.534903i	0.903241	−	0.429134i	\(-0.141181\pi\)
−0.265768	+	0.964037i	\(0.585626\pi\)
\(5\)	10.2218	−	3.72042i	0.914264	−	0.332765i	0.158310	−	0.987390i	\(-0.449396\pi\)
							0.755954	+	0.654625i	\(0.227173\pi\)
\(7\)	12.8394	−	4.67315i	0.693261	−	0.252327i	0.0287304	−	0.999587i	\(-0.490854\pi\)
							0.664531	+	0.747261i	\(0.268631\pi\)
\(11\)	16.5407	+	28.6494i	0.453383	+	0.785282i	0.998594	−	0.0530166i	\(-0.0168836\pi\)
							−0.545211	+	0.838299i	\(0.683550\pi\)
\(13\)	−4.61914	+	26.1964i	−0.0985476	+	0.558891i	0.895055	+	0.445956i	\(0.147136\pi\)
							−0.993602	+	0.112935i	\(0.963975\pi\)
\(17\)	8.03033	+	45.5422i	0.114567	+	0.649742i	0.986964	+	0.160943i	\(0.0514537\pi\)
							−0.872397	+	0.488799i	\(0.837435\pi\)
\(19\)	19.8141	+	16.6260i	0.239246	+	0.200751i	0.754525	−	0.656271i	\(-0.227867\pi\)
							−0.515279	+	0.857023i	\(0.672312\pi\)
\(23\)	21.3863	−	37.0421i	0.193885	−	0.335818i	−0.752650	−	0.658421i	\(-0.771225\pi\)
							0.946534	+	0.322603i	\(0.104558\pi\)
\(29\)	−87.7894	−	152.056i	−0.562141	−	0.973656i	−0.997309	−	0.0733074i	\(-0.976645\pi\)
							0.435169	−	0.900349i	\(-0.356689\pi\)
\(31\)	−191.093			−1.10714			−0.553569	−	0.832803i	\(-0.686735\pi\)
							−0.553569	+	0.832803i	\(0.686735\pi\)
\(37\)	−133.974	−	180.842i	−0.595275	−	0.803522i
							\(41\)	1.84706	−	10.4752i	0.00703567	−	0.0399013i	−0.981088	−	0.193563i	\(-0.937996\pi\)
0.988123	+	0.153662i	\(0.0491067\pi\)
\(43\)	−42.6629			−0.151303			−0.0756515	−	0.997134i	\(-0.524104\pi\)
							−0.0756515	+	0.997134i	\(0.524104\pi\)
\(47\)	167.460	−	290.049i	0.519714	−	0.900171i	−0.480024	−	0.877256i	\(-0.659372\pi\)
							0.999737	−	0.0229151i	\(-0.00729474\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.a.49.3	✓	24
37.34	even	9	inner	74.4.f.a.71.3	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.a.49.3	✓	24	1.1	even	1	trivial
74.4.f.a.71.3	yes	24	37.34	even	9	inner