Embedded newform 135.4.q.a.32.44

• Label: 135.4.q.a.32.44
• 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			32.44
Character	\(\chi\)	\(=\)	135.32
Dual form			135.4.q.a.38.44

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.355649 - 4.06508i) q^{2} +(-0.359649 - 5.18369i) q^{3} +(-8.51994 - 1.50230i) q^{4} +(-6.07097 + 9.38847i) q^{5} +(-21.2000 - 0.381568i) q^{6} +(-6.05376 + 8.64567i) q^{7} +(-0.687946 + 2.56745i) q^{8} +(-26.7413 + 3.72862i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.355649	−	4.06508i	0.125741	−	1.43722i	−0.627058	−	0.778973i	\(-0.715741\pi\)
							0.752799	−	0.658251i	\(-0.228703\pi\)
\(3\)	−0.359649	−	5.18369i	−0.0692146	−	0.997602i
							\(5\)	−6.07097	+	9.38847i	−0.543004	+	0.839730i
\(7\)	−6.05376	+	8.64567i	−0.326872	+	0.466822i								−0.948630	−	0.316388i	\(-0.897530\pi\)
							0.621758	+	0.783210i	\(0.286419\pi\)
\(11\)	−16.2525	+	44.6534i	−0.445483	+	1.22396i	0.490354	+	0.871523i	\(0.336868\pi\)
							−0.935837	+	0.352432i	\(0.885355\pi\)
\(13\)	−24.5480	+	2.14767i	−0.523722	+	0.0458198i	−0.345952	−	0.938252i	\(-0.612444\pi\)
							−0.177770	+	0.984072i	\(0.556888\pi\)
\(17\)	−27.7441	−	103.542i	−0.395819	−	1.47722i	−0.820380	−	0.571819i	\(-0.806238\pi\)
							0.424561	−	0.905399i	\(-0.360429\pi\)
\(19\)	−75.6711	+	43.6887i	−0.913692	+	0.527520i	−0.881617	−	0.471965i	\(-0.843545\pi\)
							−0.0320747	+	0.999485i	\(0.510211\pi\)
\(23\)	155.914	−	109.172i	1.41349	−	0.989738i	0.416755	−	0.909019i	\(-0.363167\pi\)
							0.996737	−	0.0807189i	\(-0.0257216\pi\)
\(29\)	−23.0797	+	19.3662i	−0.147786	+	0.124007i	−0.713683	−	0.700469i	\(-0.752974\pi\)
							0.565897	+	0.824476i	\(0.308530\pi\)
\(31\)	32.5604	−	184.659i	0.188646	−	1.06986i	−0.732536	−	0.680729i	\(-0.761663\pi\)
							0.921181	−	0.389134i	\(-0.127226\pi\)
\(37\)	−430.725	+	115.412i	−1.91381	+	0.512803i	−0.921608	+	0.388122i	\(0.873124\pi\)
							−0.992197	+	0.124681i	\(0.960209\pi\)
\(41\)	103.289	−	123.095i	0.393440	−	0.468883i	−0.532568	−	0.846387i	\(-0.678773\pi\)
							0.926008	+	0.377504i	\(0.123217\pi\)
\(43\)	−255.157	+	118.982i	−0.904908	+	0.421966i	−0.818640	−	0.574306i	\(-0.805272\pi\)
							−0.0862677	+	0.996272i	\(0.527494\pi\)
\(47\)	119.410	+	83.6116i	0.370590	+	0.259490i	0.744004	−	0.668175i	\(-0.232924\pi\)
							−0.373415	+	0.927665i	\(0.621813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.32.44	✓	624
5.3	odd	4	inner	135.4.q.a.113.9	yes	624
27.11	odd	18	inner	135.4.q.a.92.9	yes	624
135.38	even	36	inner	135.4.q.a.38.44	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.44	✓	624	1.1	even	1	trivial
135.4.q.a.38.44	yes	624	135.38	even	36	inner
135.4.q.a.92.9	yes	624	27.11	odd	18	inner
135.4.q.a.113.9	yes	624	5.3	odd	4	inner