Embedded newform 135.4.q.a.113.18

• Label: 135.4.q.a.113.18
• 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			113.18
Character	\(\chi\)	\(=\)	135.113
Dual form			135.4.q.a.92.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.52396 - 0.220818i) q^{2} +(3.65030 - 3.69802i) q^{3} +(-1.55687 - 0.274518i) q^{4} +(-0.664516 - 11.1606i) q^{5} +(-10.0298 + 8.52758i) q^{6} +(29.5687 + 20.7042i) q^{7} +(23.4470 + 6.28260i) q^{8} +(-0.350671 - 26.9977i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 624 q - 12 q^{2} - 12 q^{3} - 12 q^{5} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.52396	−	0.220818i	−0.892353	−	0.0780708i	−0.368251	−	0.929726i	\(-0.620043\pi\)
							−0.524102	+	0.851656i	\(0.675599\pi\)
\(3\)	3.65030	−	3.69802i	0.702500	−	0.711684i
							\(5\)	−0.664516	−	11.1606i	−0.0594362	−	0.998232i
\(7\)	29.5687	+	20.7042i	1.59656	+	1.11792i								0.926935	+	0.375222i	\(0.122434\pi\)
							0.669624	+	0.742700i	\(0.266455\pi\)
\(11\)	4.71115	−	12.9438i	0.129133	−	0.354790i	−0.858230	−	0.513266i	\(-0.828436\pi\)
							0.987363	+	0.158475i	\(0.0506578\pi\)
\(13\)	−3.28959	−	37.6002i	−0.0701822	−	0.802186i	−0.946973	−	0.321314i	\(-0.895876\pi\)
							0.876791	−	0.480872i	\(-0.159680\pi\)
\(17\)	42.8563	−	11.4833i	0.611422	−	0.163830i	0.0601969	−	0.998187i	\(-0.480827\pi\)
							0.551225	+	0.834356i	\(0.314160\pi\)
\(19\)	−59.3367	+	34.2581i	−0.716462	+	0.413649i	−0.813449	−	0.581636i	\(-0.802413\pi\)
							0.0969873	+	0.995286i	\(0.469079\pi\)
\(23\)	−68.5627	−	97.9177i	−0.621579	−	0.887706i	0.377690	−	0.925932i	\(-0.376718\pi\)
							−0.999269	+	0.0382256i	\(0.987829\pi\)
\(29\)	142.883	−	119.893i	0.914924	−	0.767712i	−0.0581258	−	0.998309i	\(-0.518512\pi\)
							0.973049	+	0.230597i	\(0.0740680\pi\)
\(31\)	16.8172	−	95.3752i	0.0974343	−	0.552577i	−0.896540	−	0.442963i	\(-0.853927\pi\)
							0.993974	−	0.109614i	\(-0.0349616\pi\)
\(37\)	53.4280	+	199.396i	0.237392	+	0.885959i	0.977056	+	0.212983i	\(0.0683178\pi\)
							−0.739664	+	0.672976i	\(0.765016\pi\)
\(41\)	34.3923	−	40.9872i	0.131004	−	0.156125i	−0.696554	−	0.717504i	\(-0.745285\pi\)
							0.827559	+	0.561379i	\(0.189729\pi\)
\(43\)	−72.8155	−	156.153i	−0.258239	−	0.553794i	0.733853	−	0.679308i	\(-0.237720\pi\)
							−0.992092	+	0.125514i	\(0.959942\pi\)
\(47\)	−180.628	+	257.964i	−0.560581	+	0.800593i	−0.994974	−	0.100131i	\(-0.968074\pi\)
							0.434393	+	0.900723i	\(0.356963\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.q.a.113.18	yes	624
5.2	odd	4	inner	135.4.q.a.32.35	✓	624
27.11	odd	18	inner	135.4.q.a.38.35	yes	624
135.92	even	36	inner	135.4.q.a.92.18	yes	624

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.q.a.32.35	✓	624	5.2	odd	4	inner
135.4.q.a.38.35	yes	624	27.11	odd	18	inner
135.4.q.a.92.18	yes	624	135.92	even	36	inner
135.4.q.a.113.18	yes	624	1.1	even	1	trivial