Embedded newform 108.4.l.a.59.13

• Label: 108.4.l.a.59.13
• Level: $108$
• Weight: $4$
• Character: 108.59
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37220628062\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(52\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.13
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.04068 + 1.95848i) q^{2} +(-2.83464 - 4.35486i) q^{3} +(0.328747 - 7.99324i) q^{4} +(-6.07198 - 16.6826i) q^{5} +(14.3135 + 3.33532i) q^{6} +(-22.8171 + 4.02326i) q^{7} +(14.9837 + 16.9555i) q^{8} +(-10.9297 + 24.6889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04068	+	1.95848i	−0.721489	+	0.692426i
							\(3\)	−2.83464	−	4.35486i	−0.545526	−	0.838094i
\(5\)	−6.07198	−	16.6826i	−0.543094	−	1.49214i								−0.842864	−	0.538127i	\(-0.819132\pi\)
							0.299769	−	0.954012i	\(-0.403090\pi\)
\(7\)	−22.8171	+	4.02326i	−1.23201	+	0.217236i	−0.751487	−	0.659748i	\(-0.770663\pi\)
							−0.480519	+	0.876984i	\(0.659552\pi\)
\(11\)	26.9709	+	9.81662i	0.739277	+	0.269075i	0.684086	−	0.729401i	\(-0.260201\pi\)
							0.0551906	+	0.998476i	\(0.482423\pi\)
\(13\)	34.3516	+	28.8244i	0.732877	+	0.614957i	0.930914	−	0.365237i	\(-0.119012\pi\)
							−0.198037	+	0.980195i	\(0.563457\pi\)
\(17\)	19.0455	+	10.9959i	0.271718	+	0.156876i	0.629668	−	0.776864i	\(-0.283191\pi\)
							−0.357950	+	0.933741i	\(0.616524\pi\)
\(19\)	−107.247	+	61.9193i	−1.29496	+	0.747645i	−0.979529	−	0.201303i	\(-0.935482\pi\)
							−0.315431	+	0.948949i	\(0.602149\pi\)
\(23\)	28.8488	−	163.610i	0.261539	−	1.48326i	−0.517174	−	0.855880i	\(-0.673016\pi\)
							0.778713	−	0.627381i	\(-0.215873\pi\)
\(29\)	19.0064	+	22.6510i	0.121704	+	0.145041i	0.823456	−	0.567380i	\(-0.192043\pi\)
							−0.701752	+	0.712421i	\(0.747599\pi\)
\(31\)	−29.1950	−	5.14787i	−0.169148	−	0.0298253i	0.0884327	−	0.996082i	\(-0.471814\pi\)
							−0.257581	+	0.966257i	\(0.582925\pi\)
\(37\)	−99.4723	+	172.291i	−0.441977	+	0.765526i	−0.997836	−	0.0657500i	\(-0.979056\pi\)
							0.555859	+	0.831276i	\(0.312389\pi\)
\(41\)	−305.793	+	364.430i	−1.16480	+	1.38816i	−0.258242	+	0.966080i	\(0.583143\pi\)
							−0.906560	+	0.422077i	\(0.861301\pi\)
\(43\)	7.64407	−	21.0019i	0.0271095	−	0.0744828i	−0.925400	−	0.378993i	\(-0.876270\pi\)
							0.952509	+	0.304510i	\(0.0984927\pi\)
\(47\)	−0.135045	−	0.765881i	−0.000419115	−	0.00237692i	0.984597	−	0.174837i	\(-0.0559397\pi\)
							−0.985017	+	0.172460i	\(0.944829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.13	yes	312
4.3	odd	2	inner	108.4.l.a.59.41	yes	312
27.11	odd	18	inner	108.4.l.a.11.41	yes	312
108.11	even	18	inner	108.4.l.a.11.13	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.13	✓	312	108.11	even	18	inner
108.4.l.a.11.41	yes	312	27.11	odd	18	inner
108.4.l.a.59.13	yes	312	1.1	even	1	trivial
108.4.l.a.59.41	yes	312	4.3	odd	2	inner