Embedded newform 135.4.q.a.113.16

• Label: 135.4.q.a.113.16
• 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			113.16
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.72407 - 0.238325i) q^{2} +(2.37154 + 4.62340i) q^{3} +(-0.514706 - 0.0907566i) q^{4} +(4.64190 + 10.1712i) q^{5} +(-5.35837 - 13.1597i) q^{6} +(7.25571 + 5.08050i) q^{7} +(22.5109 + 6.03177i) q^{8} +(-15.7516 + 21.9291i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.72407	−	0.238325i	−0.963104	−	0.0842607i	−0.405255	−	0.914203i	\(-0.632817\pi\)
							−0.557849	+	0.829943i	\(0.688373\pi\)
\(3\)	2.37154	+	4.62340i	0.456403	+	0.889773i
							\(5\)	4.64190	+	10.1712i	0.415184	+	0.909737i
\(7\)	7.25571	+	5.08050i	0.391771	+	0.274321i								0.752821	−	0.658225i	\(-0.228693\pi\)
							−0.361050	+	0.932547i	\(0.617581\pi\)
\(11\)	9.53281	−	26.1912i	0.261296	−	0.717904i	−0.737785	−	0.675036i	\(-0.764128\pi\)
							0.999081	−	0.0428681i	\(-0.0136495\pi\)
\(13\)	4.10346	+	46.9027i	0.0875457	+	1.00065i	0.905212	+	0.424960i	\(0.139712\pi\)
							−0.817667	+	0.575692i	\(0.804733\pi\)
\(17\)	−3.35507	+	0.898990i	−0.0478662	+	0.0128257i	−0.282673	−	0.959216i	\(-0.591221\pi\)
							0.234806	+	0.972042i	\(0.424554\pi\)
\(19\)	−26.6239	+	15.3713i	−0.321471	+	0.185601i	−0.652048	−	0.758178i	\(-0.726090\pi\)
							0.330577	+	0.943779i	\(0.392757\pi\)
\(23\)	−56.3824	−	80.5223i	−0.511154	−	0.730003i	0.477761	−	0.878490i	\(-0.341449\pi\)
							−0.988914	+	0.148487i	\(0.952560\pi\)
\(29\)	−96.4407	+	80.9234i	−0.617538	+	0.518176i	−0.897028	−	0.441973i	\(-0.854279\pi\)
							0.279491	+	0.960148i	\(0.409834\pi\)
\(31\)	−52.5424	+	297.983i	−0.304416	+	1.72643i	0.321826	+	0.946799i	\(0.395703\pi\)
							−0.626242	+	0.779629i	\(0.715408\pi\)
\(37\)	82.3436	+	307.310i	0.365870	+	1.36545i	0.866237	+	0.499633i	\(0.166532\pi\)
							−0.500367	+	0.865813i	\(0.666802\pi\)
\(41\)	82.6449	−	98.4923i	0.314804	−	0.375169i	−0.585320	−	0.810802i	\(-0.699031\pi\)
							0.900124	+	0.435634i	\(0.143476\pi\)
\(43\)	−179.837	−	385.661i	−0.637786	−	1.36774i	−0.913635	−	0.406536i	\(-0.866737\pi\)
							0.275848	−	0.961201i	\(-0.411041\pi\)
\(47\)	134.671	−	192.330i	0.417952	−	0.596898i	−0.553873	−	0.832601i	\(-0.686851\pi\)
							0.971825	+	0.235704i	\(0.0757395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.16	yes	624
5.2	odd	4	inner	135.4.q.a.32.37	✓	624
27.11	odd	18	inner	135.4.q.a.38.37	yes	624
135.92	even	36	inner	135.4.q.a.92.16	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.37	✓	624	5.2	odd	4	inner
135.4.q.a.38.37	yes	624	27.11	odd	18	inner
135.4.q.a.92.16	yes	624	135.92	even	36	inner
135.4.q.a.113.16	yes	624	1.1	even	1	trivial