Embedded newform 135.4.q.a.113.37

• Label: 135.4.q.a.113.37
• 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			113.37
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.43762 + 0.213264i) q^{2} +(3.56722 + 3.77822i) q^{3} +(-1.98193 - 0.349468i) q^{4} +(-9.02657 + 6.59705i) q^{5} +(7.88979 + 9.97063i) q^{6} +(16.2976 + 11.4117i) q^{7} +(-23.6651 - 6.34106i) q^{8} +(-1.54983 + 26.9555i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.43762	+	0.213264i	0.861830	+	0.0754004i	0.509495	−	0.860474i	\(-0.329832\pi\)
							0.352335	+	0.935874i	\(0.385388\pi\)
\(3\)	3.56722	+	3.77822i	0.686513	+	0.727118i
							\(5\)	−9.02657	+	6.59705i	−0.807361	+	0.590058i
\(7\)	16.2976	+	11.4117i	0.879988	+	0.616174i								0.923804	−	0.382865i	\(-0.125063\pi\)
							−0.0438160	+	0.999040i	\(0.513952\pi\)
\(11\)	2.74775	−	7.54937i	0.0753160	−	0.206929i	−0.896321	−	0.443405i	\(-0.853770\pi\)
							0.971637	+	0.236476i	\(0.0759925\pi\)
\(13\)	6.05132	+	69.1670i	0.129103	+	1.47565i	0.733892	+	0.679266i	\(0.237702\pi\)
							−0.604790	+	0.796385i	\(0.706743\pi\)
\(17\)	8.95876	−	2.40049i	0.127813	−	0.0342474i	−0.194346	−	0.980933i	\(-0.562258\pi\)
							0.322158	+	0.946686i	\(0.395592\pi\)
\(19\)	−1.99189	+	1.15002i	−0.0240511	+	0.0138859i	−0.511977	−	0.858999i	\(-0.671087\pi\)
							0.487926	+	0.872885i	\(0.337753\pi\)
\(23\)	2.50159	+	3.57264i	0.0226791	+	0.0323890i	0.830331	−	0.557270i	\(-0.188151\pi\)
							−0.807652	+	0.589659i	\(0.799262\pi\)
\(29\)	220.457	−	184.985i	1.41165	−	1.18451i	0.456011	−	0.889974i	\(-0.349278\pi\)
							0.955639	−	0.294541i	\(-0.0951667\pi\)
\(31\)	42.4700	−	240.859i	0.246059	−	1.39547i	−0.571961	−	0.820281i	\(-0.693817\pi\)
							0.818020	−	0.575190i	\(-0.195072\pi\)
\(37\)	70.6871	+	263.808i	0.314078	+	1.17216i	0.924845	+	0.380345i	\(0.124195\pi\)
							−0.610767	+	0.791810i	\(0.709139\pi\)
\(41\)	54.8377	−	65.3530i	0.208883	−	0.248937i	−0.651423	−	0.758715i	\(-0.725828\pi\)
							0.860306	+	0.509777i	\(0.170272\pi\)
\(43\)	196.948	+	422.355i	0.698470	+	1.49787i	0.859615	+	0.510943i	\(0.170704\pi\)
							−0.161144	+	0.986931i	\(0.551518\pi\)
\(47\)	−25.2063	+	35.9983i	−0.0782280	+	0.111721i	−0.856359	−	0.516381i	\(-0.827279\pi\)
							0.778131	+	0.628102i	\(0.216168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.37	yes	624
5.2	odd	4	inner	135.4.q.a.32.16	✓	624
27.11	odd	18	inner	135.4.q.a.38.16	yes	624
135.92	even	36	inner	135.4.q.a.92.37	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.16	✓	624	5.2	odd	4	inner
135.4.q.a.38.16	yes	624	27.11	odd	18	inner
135.4.q.a.92.37	yes	624	135.92	even	36	inner
135.4.q.a.113.37	yes	624	1.1	even	1	trivial