Embedded newform 74.4.f.b.53.1

• Label: 74.4.f.b.53.1
• Level: $74$
• Weight: $4$
• Character: 74.53
• 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			53.1
Character	\(\chi\)	\(=\)	74.53
Dual form			74.4.f.b.7.1

$q$-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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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	−0.895674	−	0.444711i	\(-0.853306\pi\)
−0.400272	+	0.916396i	\(0.631084\pi\)
\(5\)	−2.75980	+	15.6516i	−0.246844	+	1.39992i	0.569327	+	0.822111i	\(0.307204\pi\)
							−0.816171	+	0.577811i	\(0.803907\pi\)
\(7\)	5.01435	−	28.4378i	0.270749	−	1.53550i	−0.481399	−	0.876502i	\(-0.659871\pi\)
							0.752148	−	0.658994i	\(-0.229018\pi\)
\(11\)	−0.433038	−	0.750044i	−0.0118696	−	0.0205588i	0.860030	−	0.510244i	\(-0.170445\pi\)
							−0.871899	+	0.489685i	\(0.837112\pi\)
\(13\)	−52.8199	−	44.3212i	−1.12689	−	0.945575i	−0.127961	−	0.991779i	\(-0.540843\pi\)
							−0.998932	+	0.0462039i	\(0.985288\pi\)
\(17\)	47.0412	−	39.4723i	0.671128	−	0.563143i	−0.242271	−	0.970209i	\(-0.577892\pi\)
							0.913399	+	0.407065i	\(0.133448\pi\)
\(19\)	98.8344	−	35.9728i	1.19338	−	0.434354i	0.332469	−	0.943114i	\(-0.392118\pi\)
							0.860908	+	0.508761i	\(0.169896\pi\)
\(23\)	30.1411	−	52.2060i	0.273255	−	0.473291i	−0.696439	−	0.717616i	\(-0.745233\pi\)
							0.969693	+	0.244325i	\(0.0785665\pi\)
\(29\)	15.3095	+	26.5167i	0.0980308	+	0.169794i	0.910869	−	0.412695i	\(-0.135412\pi\)
							−0.812839	+	0.582489i	\(0.802079\pi\)
\(31\)	−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	−0.989547	−	0.144213i	\(-0.953935\pi\)
−0.313855	−	0.949471i	\(-0.601620\pi\)
\(43\)	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	−0.755721	−	0.654894i	\(-0.772713\pi\)
							0.945015	+	0.327027i	\(0.106047\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.f.b.7.1	✓	30	37.7	even	9	inner
74.4.f.b.53.1	yes	30	1.1	even	1	trivial