Embedded newform 74.4.f.b.7.5

• Label: 74.4.f.b.7.5
• Level: $74$
• Weight: $4$
• Character: 74.7
• Analytic conductor: $4.366$
• Analytic rank: $0$
• Dimension: $30$
• 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:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	74.7
Dual form			74.4.f.b.53.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87939 + 0.684040i) q^{2} +(6.95420 + 2.53112i) q^{3} +(3.06418 - 2.57115i) q^{4} +(2.65487 + 15.0565i) q^{5} -14.8010 q^{6} +(-1.00418 - 5.69497i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(21.2711 + 17.8486i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87939	+	0.684040i	−0.664463	+	0.241845i
							\(3\)	6.95420	+	2.53112i	1.33834	+	0.487115i	0.909288	−	0.416167i	\(-0.136627\pi\)
0.429048	+	0.903282i	\(0.358849\pi\)
\(5\)	2.65487	+	15.0565i	0.237459	+	1.34669i	0.837374	+	0.546631i	\(0.184090\pi\)
							−0.599915	+	0.800064i	\(0.704799\pi\)
\(7\)	−1.00418	−	5.69497i	−0.0542205	−	0.307500i	0.945622	−	0.325269i	\(-0.105455\pi\)
							−0.999842	+	0.0177690i	\(0.994344\pi\)
\(11\)	19.0761	−	33.0407i	0.522878	−	0.905651i	−0.476768	−	0.879029i	\(-0.658192\pi\)
							0.999646	−	0.0266214i	\(-0.00847486\pi\)
\(13\)	−55.8829	+	46.8913i	−1.19224	+	1.00041i	−0.192423	+	0.981312i	\(0.561635\pi\)
							−0.999818	+	0.0190968i	\(0.993921\pi\)
\(17\)	91.0273	+	76.3810i	1.29867	+	1.08971i	0.990373	+	0.138422i	\(0.0442030\pi\)
							0.308295	+	0.951291i	\(0.400241\pi\)
\(19\)	24.7038	+	8.99144i	0.298286	+	0.108567i	0.486828	−	0.873498i	\(-0.338154\pi\)
							−0.188542	+	0.982065i	\(0.560376\pi\)
\(23\)	−62.8740	−	108.901i	−0.570006	−	0.987280i	−0.996565	−	0.0828196i	\(-0.973607\pi\)
							0.426558	−	0.904460i	\(-0.359726\pi\)
\(29\)	93.1453	−	161.332i	0.596436	−	1.03306i	−0.396906	−	0.917859i	\(-0.629916\pi\)
							0.993342	−	0.115199i	\(-0.0367505\pi\)
\(31\)	76.2804			0.441947			0.220974	−	0.975280i	\(-0.429076\pi\)
							0.220974	+	0.975280i	\(0.429076\pi\)
\(37\)	−108.141	−	197.379i	−0.480492	−	0.876999i
							\(41\)	32.1784	−	27.0009i	0.122571	−	0.102850i	−0.579442	−	0.815014i	\(-0.696729\pi\)
0.702013	+	0.712164i	\(0.252285\pi\)
\(43\)	83.1778			0.294988			0.147494	−	0.989063i	\(-0.452879\pi\)
							0.147494	+	0.989063i	\(0.452879\pi\)
\(47\)	−286.373	−	496.013i	−0.888762	−	1.53938i	−0.841340	−	0.540506i	\(-0.818233\pi\)
							−0.0474213	−	0.998875i	\(-0.515100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.7.5	✓	30
37.16	even	9	inner	74.4.f.b.53.5	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.7.5	✓	30	1.1	even	1	trivial
74.4.f.b.53.5	yes	30	37.16	even	9	inner