Embedded newform 108.5.j.a.7.2

• Label: 108.5.j.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	108.7
Dual form			108.5.j.a.31.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98892 - 0.297585i) q^{2} +(-6.50989 + 6.21461i) q^{3} +(15.8229 + 2.37409i) q^{4} +(-0.917019 - 5.20068i) q^{5} +(27.8168 - 22.8523i) q^{6} +(-5.99386 + 7.14320i) q^{7} +(-62.4097 - 14.1787i) q^{8} +(3.75733 - 80.9128i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 420 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 3 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98892	−	0.297585i	−0.997229	−	0.0743963i
							\(3\)	−6.50989	+	6.21461i	−0.723321	+	0.690512i
\(5\)	−0.917019	−	5.20068i	−0.0366808	−	0.208027i								0.960959	−	0.276690i	\(-0.0892377\pi\)
							−0.997640	+	0.0686634i	\(0.978127\pi\)
\(7\)	−5.99386	+	7.14320i	−0.122324	+	0.145780i	−0.823731	−	0.566981i	\(-0.808111\pi\)
							0.701407	+	0.712761i	\(0.252556\pi\)
\(11\)	−58.8405	−	10.3752i	−0.486285	−	0.0857451i	−0.0748699	−	0.997193i	\(-0.523854\pi\)
							−0.411415	+	0.911448i	\(0.634965\pi\)
\(13\)	−86.9816	−	31.6587i	−0.514684	−	0.187330i	0.0716028	−	0.997433i	\(-0.477189\pi\)
							−0.586287	+	0.810104i	\(0.699411\pi\)
\(17\)	−114.794	−	198.828i	−0.397209	−	0.687987i	0.596171	−	0.802857i	\(-0.296688\pi\)
							−0.993380	+	0.114871i	\(0.963355\pi\)
\(19\)	504.309	+	291.163i	1.39698	+	0.806546i	0.994075	−	0.108696i	\(-0.0346676\pi\)
							0.402904	+	0.915242i	\(0.368001\pi\)
\(23\)	22.5219	+	26.8405i	0.0425745	+	0.0507383i	0.786911	−	0.617066i	\(-0.211679\pi\)
							−0.744337	+	0.667804i	\(0.767234\pi\)
\(29\)	1345.29	−	489.645i	1.59963	−	0.582218i	0.620279	−	0.784381i	\(-0.287019\pi\)
							0.979352	+	0.202163i	\(0.0647971\pi\)
\(31\)	469.928	+	560.038i	0.488999	+	0.582766i	0.952963	−	0.303088i	\(-0.0980176\pi\)
							−0.463964	+	0.885854i	\(0.653573\pi\)
\(37\)	−224.689	−	389.173i	−0.164126	−	0.284275i	0.772218	−	0.635357i	\(-0.219147\pi\)
							−0.936345	+	0.351082i	\(0.885814\pi\)
\(41\)	−510.074	−	185.652i	−0.303435	−	0.110441i	0.185815	−	0.982585i	\(-0.440507\pi\)
							−0.489250	+	0.872143i	\(0.662730\pi\)
\(43\)	1856.97	+	327.433i	1.00431	+	0.177087i	0.651532	−	0.758621i	\(-0.274127\pi\)
							0.352776	+	0.935708i	\(0.385238\pi\)
\(47\)	−216.282	+	257.755i	−0.0979095	+	0.116684i	−0.812774	−	0.582579i	\(-0.802044\pi\)
							0.714865	+	0.699263i	\(0.246488\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.j.a.7.2	✓	420
4.3	odd	2	inner	108.5.j.a.7.18	yes	420
27.4	even	9	inner	108.5.j.a.31.18	yes	420
108.31	odd	18	inner	108.5.j.a.31.2	yes	420

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.j.a.7.2	✓	420	1.1	even	1	trivial
108.5.j.a.7.18	yes	420	4.3	odd	2	inner
108.5.j.a.31.2	yes	420	108.31	odd	18	inner
108.5.j.a.31.18	yes	420	27.4	even	9	inner