Embedded newform 135.4.q.a.113.8

• Label: 135.4.q.a.113.8
• Level: $135$
• Weight: $4$
• Character: 135.113
• Analytic conductor: $7.965$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.96525785077\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(52\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			113.8
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.27250 - 0.373795i) q^{2} +(2.23339 + 4.69169i) q^{3} +(10.2360 + 1.80489i) q^{4} +(7.55455 - 8.24189i) q^{5} +(-7.78842 - 20.8801i) q^{6} +(-18.4617 - 12.9270i) q^{7} +(-9.91735 - 2.65735i) q^{8} +(-17.0239 + 20.9568i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.27250	−	0.373795i	−1.51056	−	0.132156i	−0.698447	−	0.715662i	\(-0.746125\pi\)
							−0.812109	+	0.583506i	\(0.801681\pi\)
\(3\)	2.23339	+	4.69169i	0.429816	+	0.902916i
							\(5\)	7.55455	−	8.24189i	0.675700	−	0.737177i
\(7\)	−18.4617	−	12.9270i	−0.996839	−	0.697994i								−0.0432372	−	0.999065i	\(-0.513767\pi\)
							−0.953602	+	0.301071i	\(0.902656\pi\)
\(11\)	−8.87306	+	24.3785i	−0.243212	+	0.668219i	0.756684	+	0.653781i	\(0.226818\pi\)
							−0.999896	+	0.0144380i	\(0.995404\pi\)
\(13\)	−4.52681	−	51.7416i	−0.0965777	−	1.10389i	−0.877603	−	0.479389i	\(-0.840858\pi\)
							0.781025	−	0.624500i	\(-0.214697\pi\)
\(17\)	55.5548	−	14.8859i	0.792589	−	0.212374i	0.160262	−	0.987075i	\(-0.448766\pi\)
							0.632328	+	0.774701i	\(0.282100\pi\)
\(19\)	−112.652	+	65.0398i	−1.36022	+	0.785323i	−0.989653	−	0.143482i	\(-0.954170\pi\)
							−0.370567	+	0.928806i	\(0.620837\pi\)
\(23\)	−82.8933	−	118.384i	−0.751498	−	1.07325i	−0.994654	−	0.103263i	\(-0.967072\pi\)
							0.243156	−	0.969987i	\(-0.421817\pi\)
\(29\)	−53.9182	+	45.2427i	−0.345254	+	0.289702i	−0.798881	−	0.601489i	\(-0.794574\pi\)
							0.453627	+	0.891192i	\(0.350130\pi\)
\(31\)	45.0868	−	255.700i	0.261220	−	1.48145i	−0.518366	−	0.855159i	\(-0.673459\pi\)
							0.779586	−	0.626295i	\(-0.215429\pi\)
\(37\)	−62.4562	−	233.090i	−0.277507	−	1.03567i	−0.954143	−	0.299351i	\(-0.903230\pi\)
							0.676636	−	0.736317i	\(-0.263437\pi\)
\(41\)	217.013	−	258.627i	0.826629	−	0.985139i	−0.173370	−	0.984857i	\(-0.555466\pi\)
							1.00000		0.000281980i	\(-8.97570e-5\pi\)
\(43\)	−22.7268	−	48.7377i	−0.0806000	−	0.172847i	0.861878	−	0.507116i	\(-0.169288\pi\)
							−0.942478	+	0.334269i	\(0.891511\pi\)
\(47\)	51.2721	−	73.2241i	0.159123	−	0.227252i	−0.731671	−	0.681658i	\(-0.761259\pi\)
							0.890795	+	0.454406i	\(0.150148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.8	yes	624
5.2	odd	4	inner	135.4.q.a.32.45	✓	624
27.11	odd	18	inner	135.4.q.a.38.45	yes	624
135.92	even	36	inner	135.4.q.a.92.8	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.45	✓	624	5.2	odd	4	inner
135.4.q.a.38.45	yes	624	27.11	odd	18	inner
135.4.q.a.92.8	yes	624	135.92	even	36	inner
135.4.q.a.113.8	yes	624	1.1	even	1	trivial