Embedded newform 108.5.j.a.7.11

• Label: 108.5.j.a.7.11
• Level: $108$
• Weight: $5$
• Character: 108.7
• Analytic conductor: $11.164$
• Analytic rank: $0$
• Dimension: $420$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.1639560131\)
Analytic rank:	\(0\)
Dimension:	\(420\)
Relative dimension:	\(70\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			7.11
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.47391 + 1.98292i) q^{2} +(-0.964020 - 8.94822i) q^{3} +(8.13606 - 13.7770i) q^{4} +(7.19320 + 40.7946i) q^{5} +(21.0925 + 29.1737i) q^{6} +(36.3946 - 43.3734i) q^{7} +(-0.945324 + 63.9930i) q^{8} +(-79.1413 + 17.2525i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.47391	+	1.98292i	−0.868477	+	0.495730i
							\(3\)	−0.964020	−	8.94822i	−0.107113	−	0.994247i
\(5\)	7.19320	+	40.7946i	0.287728	+	1.63179i								0.695376	+	0.718646i	\(0.255238\pi\)
							−0.407648	+	0.913139i	\(0.633651\pi\)
\(7\)	36.3946	−	43.3734i	0.742748	−	0.885172i	−0.253879	−	0.967236i	\(-0.581707\pi\)
							0.996627	+	0.0820636i	\(0.0261511\pi\)
\(11\)	−183.832	−	32.4145i	−1.51927	−	0.267888i	−0.649124	−	0.760683i	\(-0.724864\pi\)
							−0.870146	+	0.492794i	\(0.835975\pi\)
\(13\)	185.616	+	67.5587i	1.09832	+	0.399755i	0.826696	−	0.562649i	\(-0.190218\pi\)
							0.271623	+	0.962404i	\(0.412440\pi\)
\(17\)	116.506	+	201.794i	0.403135	+	0.698250i	0.994102	−	0.108445i	\(-0.0345872\pi\)
							−0.590967	+	0.806695i	\(0.701254\pi\)
\(19\)	367.397	+	212.117i	1.01772	+	0.587581i	0.913443	−	0.406968i	\(-0.133414\pi\)
							0.104277	+	0.994548i	\(0.466747\pi\)
\(23\)	209.549	+	249.730i	0.396122	+	0.472080i	0.926834	−	0.375472i	\(-0.122520\pi\)
							−0.530711	+	0.847553i	\(0.678075\pi\)
\(29\)	1348.92	−	490.965i	1.60394	−	0.583788i	0.623714	−	0.781653i	\(-0.285623\pi\)
							0.980229	+	0.197865i	\(0.0634008\pi\)
\(31\)	502.268	+	598.580i	0.522652	+	0.622872i	0.961206	−	0.275833i	\(-0.0889537\pi\)
							−0.438554	+	0.898705i	\(0.644509\pi\)
\(37\)	452.162	+	783.167i	0.330286	+	0.572073i	0.982568	−	0.185904i	\(-0.0595214\pi\)
							−0.652282	+	0.757977i	\(0.726188\pi\)
\(41\)	−646.730	−	235.391i	−0.384730	−	0.140030i	0.142412	−	0.989807i	\(-0.454514\pi\)
							−0.527142	+	0.849777i	\(0.676736\pi\)
\(43\)	873.650	+	154.048i	0.472499	+	0.0833143i	0.404827	−	0.914393i	\(-0.367332\pi\)
							0.0676718	+	0.997708i	\(0.478443\pi\)
\(47\)	489.771	−	583.687i	0.221716	−	0.264231i	−0.643708	−	0.765272i	\(-0.722605\pi\)
							0.865424	+	0.501040i	\(0.167049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.11	yes	420
4.3	odd	2	inner	108.5.j.a.7.6	✓	420
27.4	even	9	inner	108.5.j.a.31.6	yes	420
108.31	odd	18	inner	108.5.j.a.31.11	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.6	✓	420	4.3	odd	2	inner
108.5.j.a.7.11	yes	420	1.1	even	1	trivial
108.5.j.a.31.6	yes	420	27.4	even	9	inner
108.5.j.a.31.11	yes	420	108.31	odd	18	inner