Embedded newform 135.4.q.a.113.1

• Label: 135.4.q.a.113.1
• 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			113.1
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.28656 - 0.462514i) q^{2} +(-5.01597 - 1.35649i) q^{3} +(19.8553 + 3.50103i) q^{4} +(-6.33733 - 9.21077i) q^{5} +(25.8898 + 9.49113i) q^{6} +(1.80220 + 1.26191i) q^{7} +(-62.3397 - 16.7039i) q^{8} +(23.3199 + 13.6082i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.28656	−	0.462514i	−1.86908	−	0.163523i	−0.904356	−	0.426780i	\(-0.859648\pi\)
							−0.964726	+	0.263256i	\(0.915203\pi\)
\(3\)	−5.01597	−	1.35649i	−0.965323	−	0.261057i
							\(5\)	−6.33733	−	9.21077i	−0.566828	−	0.823836i
\(7\)	1.80220	+	1.26191i	0.0973095	+	0.0681368i								0.621219	−	0.783637i	\(-0.286638\pi\)
							−0.523909	+	0.851774i	\(0.675527\pi\)
\(11\)	−20.8521	+	57.2906i	−0.571558	+	1.57034i	0.230485	+	0.973076i	\(0.425969\pi\)
							−0.802043	+	0.597266i	\(0.796254\pi\)
\(13\)	1.38793	+	15.8641i	0.0296109	+	0.338454i	0.996521	+	0.0833380i	\(0.0265581\pi\)
							−0.966910	+	0.255116i	\(0.917886\pi\)
\(17\)	50.1216	−	13.4300i	0.715074	−	0.191604i	0.117102	−	0.993120i	\(-0.462640\pi\)
							0.597973	+	0.801516i	\(0.295973\pi\)
\(19\)	−11.0244	+	6.36496i	−0.133115	+	0.0768537i	−0.565078	−	0.825037i	\(-0.691154\pi\)
							0.431964	+	0.901891i	\(0.357821\pi\)
\(23\)	−35.1565	−	50.2087i	−0.318723	−	0.455184i	0.627538	−	0.778586i	\(-0.284063\pi\)
							−0.946262	+	0.323401i	\(0.895174\pi\)
\(29\)	185.729	−	155.845i	1.18927	−	0.997919i	0.189402	−	0.981900i	\(-0.439345\pi\)
							0.999872	−	0.0160190i	\(-0.00509923\pi\)
\(31\)	52.9226	−	300.139i	0.306619	−	1.73892i	−0.309166	−	0.951008i	\(-0.600050\pi\)
							0.615785	−	0.787914i	\(-0.288839\pi\)
\(37\)	100.871	+	376.456i	0.448192	+	1.67268i	0.707368	+	0.706845i	\(0.249882\pi\)
							−0.259176	+	0.965830i	\(0.583451\pi\)
\(41\)	−6.64978	+	7.92490i	−0.0253298	+	0.0301869i	−0.778561	−	0.627569i	\(-0.784050\pi\)
							0.753231	+	0.657756i	\(0.228494\pi\)
\(43\)	−75.3944	−	161.684i	−0.267385	−	0.573408i	0.726072	−	0.687619i	\(-0.241344\pi\)
							−0.993457	+	0.114210i	\(0.963566\pi\)
\(47\)	−151.588	+	216.491i	−0.470456	+	0.671881i	−0.982325	−	0.187183i	\(-0.940064\pi\)
							0.511869	+	0.859063i	\(0.328953\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.1	yes	624
5.2	odd	4	inner	135.4.q.a.32.52	✓	624
27.11	odd	18	inner	135.4.q.a.38.52	yes	624
135.92	even	36	inner	135.4.q.a.92.1	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.52	✓	624	5.2	odd	4	inner
135.4.q.a.38.52	yes	624	27.11	odd	18	inner
135.4.q.a.92.1	yes	624	135.92	even	36	inner
135.4.q.a.113.1	yes	624	1.1	even	1	trivial