Embedded newform 135.4.p.a.4.2

• Label: 135.4.p.a.4.2
• Level: $135$
• Weight: $4$
• Character: 135.4
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.p (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
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			4.2
Character	\(\chi\)	\(=\)	135.4
Dual form			135.4.p.a.34.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85666 + 5.10114i) q^{2} +(-2.41243 + 4.60219i) q^{3} +(-16.4461 - 13.7999i) q^{4} +(4.19064 + 10.3653i) q^{5} +(-18.9974 - 20.8509i) q^{6} +(-9.50379 - 11.3262i) q^{7} +(63.3201 - 36.5579i) q^{8} +(-15.3604 - 22.2049i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 12 q^{4} - 18 q^{5} - 24 q^{6} + 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85666	+	5.10114i	−0.656430	+	1.80353i	−0.0639303	+	0.997954i	\(0.520364\pi\)
							−0.592499	+	0.805571i	\(0.701859\pi\)
\(3\)	−2.41243	+	4.60219i	−0.464272	+	0.885692i
							\(5\)	4.19064	+	10.3653i	0.374822	+	0.927097i
\(7\)	−9.50379	−	11.3262i	−0.513157	−	0.611556i								0.445792	−	0.895137i	\(-0.352922\pi\)
							−0.958948	+	0.283580i	\(0.908478\pi\)
\(11\)	−4.28731	−	24.3146i	−0.117516	−	0.666465i	−0.985474	−	0.169828i	\(-0.945679\pi\)
							0.867958	−	0.496638i	\(-0.165432\pi\)
\(13\)	13.9967	+	38.4556i	0.298614	+	0.820436i	0.994732	+	0.102509i	\(0.0326869\pi\)
							−0.696118	+	0.717928i	\(0.745091\pi\)
\(17\)	−23.4603	−	13.5448i	−0.334704	−	0.193241i	0.323224	−	0.946323i	\(-0.395233\pi\)
							−0.657928	+	0.753081i	\(0.728567\pi\)
\(19\)	−65.1377	−	112.822i	−0.786506	−	1.36227i	−0.928095	−	0.372343i	\(-0.878554\pi\)
							0.141589	−	0.989926i	\(-0.454779\pi\)
\(23\)	−95.6641	+	114.008i	−0.867276	+	1.03358i	0.131829	+	0.991273i	\(0.457915\pi\)
							−0.999105	+	0.0423067i	\(0.986529\pi\)
\(29\)	152.273	+	55.4227i	0.975046	+	0.354888i	0.779912	−	0.625889i	\(-0.215264\pi\)
							0.195134	+	0.980777i	\(0.437486\pi\)
\(31\)	−10.8508	−	9.10494i	−0.0628668	−	0.0527515i	0.610813	−	0.791775i	\(-0.290843\pi\)
							−0.673680	+	0.739024i	\(0.735287\pi\)
\(37\)	−239.445	−	138.244i	−1.06391	−	0.614246i	−0.137396	−	0.990516i	\(-0.543873\pi\)
							−0.926510	+	0.376270i	\(0.877207\pi\)
\(41\)	−25.4032	+	9.24601i	−0.0967638	+	0.0352191i	−0.389948	−	0.920837i	\(-0.627507\pi\)
							0.293185	+	0.956056i	\(0.405285\pi\)
\(43\)	299.774	−	52.8583i	1.06314	−	0.187461i	0.385392	−	0.922753i	\(-0.374066\pi\)
							0.677750	+	0.735292i	\(0.262955\pi\)
\(47\)	−311.417	−	371.133i	−0.966487	−	1.15181i	−0.988372	−	0.152054i	\(-0.951411\pi\)
							0.0218851	−	0.999760i	\(-0.493033\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.p.a.4.2	✓	312
5.4	even	2	inner	135.4.p.a.4.51	yes	312
27.7	even	9	inner	135.4.p.a.34.51	yes	312
135.34	even	18	inner	135.4.p.a.34.2	yes	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.p.a.4.2	✓	312	1.1	even	1	trivial
135.4.p.a.4.51	yes	312	5.4	even	2	inner
135.4.p.a.34.2	yes	312	135.34	even	18	inner
135.4.p.a.34.51	yes	312	27.7	even	9	inner