Embedded newform 74.4.f.b.7.1

• Label: 74.4.f.b.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	74.7
Dual form			74.4.f.b.53.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87939 + 0.684040i) q^{2} +(-6.73393 - 2.45095i) q^{3} +(3.06418 - 2.57115i) q^{4} +(-2.75980 - 15.6516i) q^{5} +14.3322 q^{6} +(5.01435 + 28.4378i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(18.6555 + 15.6538i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} - 6 q^{5} - 36 q^{6} - 75 q^{7} - 120 q^{8} - 48 q^{9}+O(q^{10}) \)

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
							\(3\)	−6.73393	−	2.45095i	−1.29595	−	0.471686i	−0.400272	−	0.916396i	\(-0.631084\pi\)
−0.895674	+	0.444711i	\(0.853306\pi\)
\(5\)	−2.75980	−	15.6516i	−0.246844	−	1.39992i	−0.816171	−	0.577811i	\(-0.803907\pi\)
							0.569327	−	0.822111i	\(-0.307204\pi\)
\(7\)	5.01435	+	28.4378i	0.270749	+	1.53550i	0.752148	+	0.658994i	\(0.229018\pi\)
							−0.481399	+	0.876502i	\(0.659871\pi\)
\(11\)	−0.433038	+	0.750044i	−0.0118696	+	0.0205588i	−0.871899	−	0.489685i	\(-0.837112\pi\)
							0.860030	+	0.510244i	\(0.170445\pi\)
\(13\)	−52.8199	+	44.3212i	−1.12689	+	0.945575i	−0.998932	−	0.0462039i	\(-0.985288\pi\)
							−0.127961	+	0.991779i	\(0.540843\pi\)
\(17\)	47.0412	+	39.4723i	0.671128	+	0.563143i	0.913399	−	0.407065i	\(-0.133448\pi\)
							−0.242271	+	0.970209i	\(0.577892\pi\)
\(19\)	98.8344	+	35.9728i	1.19338	+	0.434354i	0.860908	−	0.508761i	\(-0.169896\pi\)
							0.332469	+	0.943114i	\(0.392118\pi\)
\(23\)	30.1411	+	52.2060i	0.273255	+	0.473291i	0.969693	−	0.244325i	\(-0.0785665\pi\)
							−0.696439	+	0.717616i	\(0.745233\pi\)
\(29\)	15.3095	−	26.5167i	0.0980308	−	0.169794i	−0.812839	−	0.582489i	\(-0.802079\pi\)
							0.910869	+	0.412695i	\(0.135412\pi\)
\(31\)	−322.240			−1.86697			−0.933485	−	0.358616i	\(-0.883249\pi\)
							−0.933485	+	0.358616i	\(0.883249\pi\)
\(37\)	−167.535	−	150.283i	−0.744396	−	0.667739i
							\(41\)	−342.179	+	287.123i	−1.30340	+	1.09368i	−0.313855	+	0.949471i	\(0.601620\pi\)
−0.989547	+	0.144213i	\(0.953935\pi\)
\(43\)	174.447			0.618671			0.309336	−	0.950953i	\(-0.399893\pi\)
							0.309336	+	0.950953i	\(0.399893\pi\)
\(47\)	60.9935	+	105.644i	0.189294	+	0.327867i	0.945015	−	0.327027i	\(-0.106047\pi\)
							−0.755721	+	0.654894i	\(0.772713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.f.b.7.1	✓	30
37.16	even	9	inner	74.4.f.b.53.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.7.1	✓	30	1.1	even	1	trivial
74.4.f.b.53.1	yes	30	37.16	even	9	inner