Embedded newform 74.4.f.b.9.1

• Label: 74.4.f.b.9.1
• Level: $74$
• Weight: $4$
• Character: 74.9
• 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			9.1
Character	\(\chi\)	\(=\)	74.9
Dual form			74.4.f.b.33.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 + 1.96962i) q^{2} +(-1.62017 + 9.18845i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-9.08898 - 7.62656i) q^{5} -18.6604 q^{6} +(21.4230 + 17.9761i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(-56.4309 - 20.5392i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−1.62017	+	9.18845i	−0.311802	+	1.76832i	0.277815	+	0.960635i	\(0.410390\pi\)
−0.589617	+	0.807683i	\(0.700721\pi\)
\(5\)	−9.08898	−	7.62656i	−0.812943	−	0.682140i	0.138365	−	0.990381i	\(-0.455815\pi\)
							−0.951308	+	0.308241i	\(0.900260\pi\)
\(7\)	21.4230	+	17.9761i	1.15674	+	0.970616i	0.999855	−	0.0170015i	\(-0.00541201\pi\)
							0.156880	+	0.987618i	\(0.449856\pi\)
\(11\)	−4.89440	−	8.47735i	−0.134156	−	0.232365i	0.791119	−	0.611663i	\(-0.209499\pi\)
							−0.925275	+	0.379298i	\(0.876166\pi\)
\(13\)	20.4561	−	7.44543i	0.436424	−	0.158845i	−0.114459	−	0.993428i	\(-0.536513\pi\)
							0.550883	+	0.834583i	\(0.314291\pi\)
\(17\)	−51.5252	−	18.7536i	−0.735100	−	0.267555i	−0.0527780	−	0.998606i	\(-0.516808\pi\)
							−0.682322	+	0.731052i	\(0.739030\pi\)
\(19\)	−28.0212	+	158.916i	−0.338342	+	1.91883i	0.0530165	+	0.998594i	\(0.483116\pi\)
							−0.391358	+	0.920238i	\(0.627995\pi\)
\(23\)	−56.1041	+	97.1751i	−0.508631	+	0.880974i	0.491319	+	0.870980i	\(0.336515\pi\)
							−0.999950	+	0.00999475i	\(0.996819\pi\)
\(29\)	90.1306	+	156.111i	0.577132	+	0.999622i	0.995806	+	0.0914855i	\(0.0291615\pi\)
							−0.418674	+	0.908136i	\(0.637505\pi\)
\(31\)	162.666			0.942440			0.471220	−	0.882016i	\(-0.343814\pi\)
							0.471220	+	0.882016i	\(0.343814\pi\)
\(37\)	−151.194	+	166.714i	−0.671786	+	0.740745i
							\(41\)	296.959	−	108.084i	1.13115	−	0.411706i	0.292442	−	0.956283i	\(-0.405532\pi\)
0.838711	+	0.544577i	\(0.183310\pi\)
\(43\)	−183.318			−0.650135			−0.325067	−	0.945691i	\(-0.605387\pi\)
							−0.325067	+	0.945691i	\(0.605387\pi\)
\(47\)	65.0662	−	112.698i	0.201933	−	0.349759i	−0.747218	−	0.664579i	\(-0.768611\pi\)
							0.949151	+	0.314820i	\(0.101944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.9.1	✓	30
37.33	even	9	inner	74.4.f.b.33.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.9.1	✓	30	1.1	even	1	trivial
74.4.f.b.33.1	yes	30	37.33	even	9	inner