Embedded newform 74.4.f.b.9.5

• Label: 74.4.f.b.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	74.9
Dual form			74.4.f.b.33.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 + 1.96962i) q^{2} +(1.33010 - 7.54337i) q^{3} +(-3.75877 + 1.36808i) q^{4} +(-4.89924 - 4.11095i) q^{5} +15.3195 q^{6} +(-6.34775 - 5.32640i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(-29.7615 - 10.8323i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

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
							\(3\)	1.33010	−	7.54337i	0.255978	−	1.45172i	−0.537573	−	0.843217i	\(-0.680659\pi\)
0.793551	−	0.608504i	\(-0.208230\pi\)
\(5\)	−4.89924	−	4.11095i	−0.438202	−	0.367695i	0.396834	−	0.917890i	\(-0.370109\pi\)
							−0.835036	+	0.550196i	\(0.814553\pi\)
\(7\)	−6.34775	−	5.32640i	−0.342746	−	0.287598i	0.455123	−	0.890428i	\(-0.349595\pi\)
							−0.797870	+	0.602830i	\(0.794040\pi\)
\(11\)	−32.5912	−	56.4496i	−0.893328	−	1.54729i	−0.835860	−	0.548942i	\(-0.815031\pi\)
							−0.0574679	−	0.998347i	\(-0.518303\pi\)
\(13\)	24.8424	−	9.04190i	0.530004	−	0.192906i	−0.0631363	−	0.998005i	\(-0.520110\pi\)
							0.593140	+	0.805099i	\(0.297888\pi\)
\(17\)	96.1207	+	34.9851i	1.37133	+	0.499125i	0.919540	−	0.392997i	\(-0.128562\pi\)
							0.451795	+	0.892122i	\(0.350784\pi\)
\(19\)	−11.5967	+	65.7679i	−0.140024	+	0.794115i	0.831205	+	0.555966i	\(0.187652\pi\)
							−0.971229	+	0.238149i	\(0.923459\pi\)
\(23\)	41.4085	−	71.7217i	0.375403	−	0.650218i	−0.614984	−	0.788540i	\(-0.710838\pi\)
							0.990387	+	0.138322i	\(0.0441708\pi\)
\(29\)	113.346	+	196.321i	0.725788	+	1.25710i	0.958649	+	0.284592i	\(0.0918582\pi\)
							−0.232860	+	0.972510i	\(0.574808\pi\)
\(31\)	−94.8681			−0.549639			−0.274820	−	0.961496i	\(-0.588618\pi\)
							−0.274820	+	0.961496i	\(0.588618\pi\)
\(37\)	93.2459	+	204.837i	0.414312	+	0.910135i
							\(41\)	−37.3117	+	13.5803i	−0.142125	+	0.0517291i	−0.412103	−	0.911137i	\(-0.635206\pi\)
0.269978	+	0.962866i	\(0.412983\pi\)
\(43\)	−48.1559			−0.170784			−0.0853920	−	0.996347i	\(-0.527214\pi\)
							−0.0853920	+	0.996347i	\(0.527214\pi\)
\(47\)	95.8190	−	165.963i	0.297375	−	0.515069i	−0.678159	−	0.734915i	\(-0.737222\pi\)
							0.975535	+	0.219846i	\(0.0705554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.9.5	✓	30
37.33	even	9	inner	74.4.f.b.33.5	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.9.5	✓	30	1.1	even	1	trivial
74.4.f.b.33.5	yes	30	37.33	even	9	inner