Embedded newform 135.4.q.a.113.3

• Label: 135.4.q.a.113.3
• 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			113.3
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.21804 - 0.456519i) q^{2} +(1.63923 - 4.93081i) q^{3} +(19.1411 + 3.37509i) q^{4} +(-2.66423 + 10.8583i) q^{5} +(-10.8046 + 24.9808i) q^{6} +(7.48250 + 5.23930i) q^{7} +(-57.8622 - 15.5041i) q^{8} +(-21.6258 - 16.1655i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.21804	−	0.456519i	−1.84486	−	0.161404i	−0.889514	−	0.456907i	\(-0.848957\pi\)
							−0.955342	+	0.295503i	\(0.904513\pi\)
\(3\)	1.63923	−	4.93081i	0.315471	−	0.948935i
							\(5\)	−2.66423	+	10.8583i	−0.238296	+	0.971193i
\(7\)	7.48250	+	5.23930i	0.404017	+	0.282896i								0.757870	−	0.652405i	\(-0.226240\pi\)
							−0.353853	+	0.935301i	\(0.615129\pi\)
\(11\)	−8.10175	+	22.2594i	−0.222070	+	0.610132i	−0.999830	−	0.0184288i	\(-0.994134\pi\)
							0.777760	+	0.628561i	\(0.216356\pi\)
\(13\)	−5.04795	−	57.6983i	−0.107696	−	1.23097i	−0.837162	−	0.546955i	\(-0.815787\pi\)
							0.729466	−	0.684017i	\(-0.239769\pi\)
\(17\)	−10.4057	+	2.78819i	−0.148455	+	0.0397785i	−0.332281	−	0.943180i	\(-0.607818\pi\)
							0.183826	+	0.982959i	\(0.441152\pi\)
\(19\)	−73.3773	+	42.3644i	−0.885996	+	0.511530i	−0.872631	−	0.488381i	\(-0.837588\pi\)
							−0.0133651	+	0.999911i	\(0.504254\pi\)
\(23\)	−107.230	−	153.140i	−0.972131	−	1.38835i	−0.920960	−	0.389657i	\(-0.872594\pi\)
							−0.0511709	−	0.998690i	\(-0.516295\pi\)
\(29\)	−104.075	+	87.3291i	−0.666421	+	0.559193i	−0.912004	−	0.410182i	\(-0.865465\pi\)
							0.245583	+	0.969376i	\(0.421021\pi\)
\(31\)	−49.2441	+	279.277i	−0.285306	+	1.61805i	0.418883	+	0.908040i	\(0.362422\pi\)
							−0.704189	+	0.710013i	\(0.748689\pi\)
\(37\)	34.4825	+	128.691i	0.153213	+	0.571800i	0.999252	+	0.0386780i	\(0.0123147\pi\)
							−0.846038	+	0.533122i	\(0.821019\pi\)
\(41\)	28.3022	−	33.7292i	0.107806	−	0.128478i	−0.709443	−	0.704763i	\(-0.751053\pi\)
							0.817249	+	0.576284i	\(0.195498\pi\)
\(43\)	2.04407	+	4.38351i	0.00724924	+	0.0155460i	0.909899	−	0.414830i	\(-0.136159\pi\)
							−0.902650	+	0.430376i	\(0.858381\pi\)
\(47\)	−46.5813	+	66.5250i	−0.144566	+	0.206461i	−0.884904	−	0.465773i	\(-0.845776\pi\)
							0.740339	+	0.672234i	\(0.234665\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.3	yes	624
5.2	odd	4	inner	135.4.q.a.32.50	✓	624
27.11	odd	18	inner	135.4.q.a.38.50	yes	624
135.92	even	36	inner	135.4.q.a.92.3	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.50	✓	624	5.2	odd	4	inner
135.4.q.a.38.50	yes	624	27.11	odd	18	inner
135.4.q.a.92.3	yes	624	135.92	even	36	inner
135.4.q.a.113.3	yes	624	1.1	even	1	trivial