Embedded newform 135.4.q.a.32.31

• Label: 135.4.q.a.32.31
• Level: $135$
• Weight: $4$
• Character: 135.32
• 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			32.31
Character	\(\chi\)	\(=\)	135.32
Dual form			135.4.q.a.38.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0969803 - 1.10849i) q^{2} +(3.93052 + 3.39868i) q^{3} +(6.65912 + 1.17418i) q^{4} +(-7.85662 + 7.95446i) q^{5} +(4.14859 - 4.02734i) q^{6} +(16.5158 - 23.5869i) q^{7} +(4.25133 - 15.8662i) q^{8} +(3.89795 + 26.7171i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0969803	−	1.10849i	0.0342877	−	0.391911i	−0.959588	−	0.281408i	\(-0.909198\pi\)
							0.993876	−	0.110502i	\(-0.0352460\pi\)
\(3\)	3.93052	+	3.39868i	0.756429	+	0.654076i
							\(5\)	−7.85662	+	7.95446i	−0.702718	+	0.711469i
\(7\)	16.5158	−	23.5869i	0.891767	−	1.27358i								−0.0693948	−	0.997589i	\(-0.522107\pi\)
							0.961162	−	0.275986i	\(-0.0890043\pi\)
\(11\)	−2.37318	+	6.52025i	−0.0650491	+	0.178721i	−0.967959	−	0.251110i	\(-0.919204\pi\)
							0.902910	+	0.429831i	\(0.141427\pi\)
\(13\)	51.9243	−	4.54279i	1.10779	−	0.0969187i	0.481451	−	0.876473i	\(-0.340110\pi\)
							0.626334	+	0.779554i	\(0.284554\pi\)
\(17\)	24.9662	+	93.1752i	0.356188	+	1.32931i	0.878983	+	0.476853i	\(0.158223\pi\)
							−0.522795	+	0.852458i	\(0.675111\pi\)
\(19\)	−107.354	+	61.9811i	−1.29625	+	0.748391i	−0.979755	−	0.200202i	\(-0.935840\pi\)
							−0.316497	+	0.948593i	\(0.602507\pi\)
\(23\)	−4.27929	+	2.99639i	−0.0387954	+	0.0271648i	−0.592813	−	0.805340i	\(-0.701983\pi\)
							0.554018	+	0.832505i	\(0.313094\pi\)
\(29\)	48.2501	−	40.4867i	0.308959	−	0.259248i	−0.475102	−	0.879931i	\(-0.657589\pi\)
							0.784062	+	0.620683i	\(0.213145\pi\)
\(31\)	41.3524	−	234.521i	0.239584	−	1.35875i	−0.593157	−	0.805087i	\(-0.702119\pi\)
							0.832741	−	0.553662i	\(-0.186770\pi\)
\(37\)	−266.348	+	71.3678i	−1.18344	+	0.317103i	−0.796291	−	0.604913i	\(-0.793208\pi\)
							−0.387152	+	0.922016i	\(0.626541\pi\)
\(41\)	166.426	−	198.339i	0.633937	−	0.755497i	−0.349462	−	0.936950i	\(-0.613636\pi\)
							0.983400	+	0.181454i	\(0.0580802\pi\)
\(43\)	−54.1896	+	25.2690i	−0.192182	+	0.0896160i	−0.516327	−	0.856391i	\(-0.672701\pi\)
							0.324145	+	0.946007i	\(0.394923\pi\)
\(47\)	−156.527	−	109.602i	−0.485785	−	0.340150i	0.304889	−	0.952388i	\(-0.401381\pi\)
							−0.790674	+	0.612238i	\(0.790270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.32.31	✓	624
5.3	odd	4	inner	135.4.q.a.113.22	yes	624
27.11	odd	18	inner	135.4.q.a.92.22	yes	624
135.38	even	36	inner	135.4.q.a.38.31	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.31	✓	624	1.1	even	1	trivial
135.4.q.a.38.31	yes	624	135.38	even	36	inner
135.4.q.a.92.22	yes	624	27.11	odd	18	inner
135.4.q.a.113.22	yes	624	5.3	odd	4	inner