Embedded newform 135.2.q.a.38.8

• Label: 135.2.q.a.38.8
• Level: $135$
• Weight: $2$
• Character: 135.38
• Analytic conductor: $1.078$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.q (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.07798042729\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(16\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			38.8
Character	\(\chi\)	\(=\)	135.38
Dual form			135.2.q.a.32.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00133058 + 0.0152086i) q^{2} +(-1.71244 + 0.259886i) q^{3} +(1.96939 - 0.347256i) q^{4} +(0.516124 + 2.17569i) q^{5} +(-0.00623106 - 0.0256981i) q^{6} +(-0.148030 - 0.211408i) q^{7} +(0.0158044 + 0.0589827i) q^{8} +(2.86492 - 0.890079i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 36 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00133058	+	0.0152086i	0.000940864	+	0.0107541i	0.996651	−	0.0817774i	\(-0.0260597\pi\)
							−0.995710	+	0.0925315i	\(0.970504\pi\)
\(3\)	−1.71244	+	0.259886i	−0.988679	+	0.150045i
							\(5\)	0.516124	+	2.17569i	0.230818	+	0.972997i
\(7\)	−0.148030	−	0.211408i	−0.0559499	−	0.0799048i								0.790202	−	0.612846i	\(-0.209975\pi\)
							−0.846152	+	0.532941i	\(0.821087\pi\)
\(11\)	1.49616	+	4.11066i	0.451108	+	1.23941i	0.931945	+	0.362600i	\(0.118111\pi\)
							−0.480836	+	0.876810i	\(0.659667\pi\)
\(13\)	2.14258	+	0.187451i	0.594244	+	0.0519896i	0.380311	−	0.924859i	\(-0.375817\pi\)
							0.213933	+	0.976848i	\(0.431373\pi\)
\(17\)	0.456091	−	1.70216i	0.110618	−	0.412833i	−0.888303	−	0.459257i	\(-0.848116\pi\)
							0.998922	+	0.0464236i	\(0.0147824\pi\)
\(19\)	−4.91894	−	2.83995i	−1.12848	−	0.651529i	−0.184929	−	0.982752i	\(-0.559206\pi\)
							−0.943553	+	0.331222i	\(0.892539\pi\)
\(23\)	−4.74660	−	3.32360i	−0.989734	−	0.693019i	−0.0377879	−	0.999286i	\(-0.512031\pi\)
							−0.951946	+	0.306267i	\(0.900920\pi\)
\(29\)	−3.59567	−	3.01712i	−0.667699	−	0.560266i	0.244685	−	0.969603i	\(-0.421316\pi\)
							−0.912383	+	0.409337i	\(0.865760\pi\)
\(31\)	−0.912568	−	5.17543i	−0.163902	−	0.929534i	−0.950189	−	0.311673i	\(-0.899111\pi\)
							0.786287	−	0.617861i	\(-0.212001\pi\)
\(37\)	3.12552	+	0.837479i	0.513832	+	0.137681i	0.506412	−	0.862292i	\(-0.330972\pi\)
							0.00741968	+	0.999972i	\(0.497638\pi\)
\(41\)	0.241984	+	0.288386i	0.0377916	+	0.0450383i	0.784609	−	0.619991i	\(-0.212864\pi\)
							−0.746818	+	0.665029i	\(0.768419\pi\)
\(43\)	8.25395	+	3.84888i	1.25872	+	0.586948i	0.933461	−	0.358678i	\(-0.116772\pi\)
							0.325254	+	0.945627i	\(0.394550\pi\)
\(47\)	−3.28345	+	2.29910i	−0.478941	+	0.335358i	−0.787990	−	0.615688i	\(-0.788878\pi\)
							0.309049	+	0.951046i	\(0.399989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.2.q.a.38.8	yes	192
3.2	odd	2		405.2.r.a.278.9		192
5.2	odd	4	inner	135.2.q.a.92.9	yes	192
5.3	odd	4		675.2.ba.b.632.8		192
5.4	even	2		675.2.ba.b.443.9		192
15.2	even	4		405.2.r.a.197.8		192
27.5	odd	18	inner	135.2.q.a.113.9	yes	192
27.22	even	9		405.2.r.a.368.8		192
135.22	odd	36		405.2.r.a.287.9		192
135.32	even	36	inner	135.2.q.a.32.8	✓	192
135.59	odd	18		675.2.ba.b.518.8		192
135.113	even	36		675.2.ba.b.32.9		192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.2.q.a.32.8	✓	192	135.32	even	36	inner
135.2.q.a.38.8	yes	192	1.1	even	1	trivial
135.2.q.a.92.9	yes	192	5.2	odd	4	inner
135.2.q.a.113.9	yes	192	27.5	odd	18	inner
405.2.r.a.197.8		192	15.2	even	4
405.2.r.a.278.9		192	3.2	odd	2
405.2.r.a.287.9		192	135.22	odd	36
405.2.r.a.368.8		192	27.22	even	9
675.2.ba.b.32.9		192	135.113	even	36
675.2.ba.b.443.9		192	5.4	even	2
675.2.ba.b.518.8		192	135.59	odd	18
675.2.ba.b.632.8		192	5.3	odd	4