Embedded newform 162.4.g.b.7.11

• Label: 162.4.g.b.7.11
• Level: $162$
• Weight: $4$
• Character: 162.7
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(14\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			7.11
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.b.139.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19432 - 1.60425i) q^{2} +(3.30686 - 4.00807i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(18.1458 - 11.9347i) q^{5} +(-10.3794 - 0.518112i) q^{6} +(-8.00040 + 8.47993i) q^{7} +(7.51754 - 2.73616i) q^{8} +(-5.12931 - 26.5083i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{8}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19432	−	1.60425i	−0.422255	−	0.567187i
							\(3\)	3.30686	−	4.00807i	0.636406	−	0.771354i
\(5\)	18.1458	−	11.9347i	1.62301	−	1.06747i								0.680795	−	0.732474i	\(-0.261634\pi\)
							0.942219	−	0.334999i	\(-0.108736\pi\)
\(7\)	−8.00040	+	8.47993i	−0.431981	+	0.457873i	−0.906396	−	0.422428i	\(-0.861178\pi\)
							0.474416	+	0.880301i	\(0.342659\pi\)
\(11\)	59.1609	−	29.7117i	1.62161	−	0.814402i	0.622103	−	0.782935i	\(-0.286278\pi\)
							0.999505	−	0.0314668i	\(-0.0100179\pi\)
\(13\)	16.5212	+	38.3004i	0.352473	+	0.817124i	0.998549	+	0.0538513i	\(0.0171497\pi\)
							−0.646076	+	0.763273i	\(0.723591\pi\)
\(17\)	−66.7600	+	56.0183i	−0.952451	+	0.799202i	−0.979709	−	0.200427i	\(-0.935767\pi\)
							0.0272572	+	0.999628i	\(0.491323\pi\)
\(19\)	43.6501	+	36.6267i	0.527053	+	0.442250i	0.867082	−	0.498165i	\(-0.165993\pi\)
							−0.340029	+	0.940415i	\(0.610437\pi\)
\(23\)	−14.2449	−	15.0988i	−0.129142	−	0.136883i	0.659575	−	0.751638i	\(-0.270736\pi\)
							−0.788718	+	0.614756i	\(0.789255\pi\)
\(29\)	−104.238	+	12.1837i	−0.667468	+	0.0780158i	−0.443077	−	0.896484i	\(-0.646113\pi\)
							−0.224391	+	0.974499i	\(0.572039\pi\)
\(31\)	−243.469	+	57.7032i	−1.41059	+	0.334316i	−0.864172	−	0.503196i	\(-0.832157\pi\)
							−0.546419	+	0.837512i	\(0.684009\pi\)
\(37\)	−52.7099	−	298.932i	−0.234201	−	1.32822i	−0.844289	−	0.535888i	\(-0.819977\pi\)
							0.610088	−	0.792334i	\(-0.291134\pi\)
\(41\)	−11.4028	+	15.3167i	−0.0434347	+	0.0583429i	−0.823310	−	0.567592i	\(-0.807875\pi\)
							0.779875	+	0.625935i	\(0.215282\pi\)
\(43\)	−21.4524	+	368.324i	−0.0760805	+	1.30625i	0.716973	+	0.697101i	\(0.245527\pi\)
							−0.793054	+	0.609152i	\(0.791510\pi\)
\(47\)	−126.118	−	29.8906i	−0.391410	−	0.0927659i	0.0301983	−	0.999544i	\(-0.490386\pi\)
							−0.421608	+	0.906778i	\(0.638534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.7.11	✓	252
81.58	even	27	inner	162.4.g.b.139.11	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.7.11	✓	252	1.1	even	1	trivial
162.4.g.b.139.11	yes	252	81.58	even	27	inner