Embedded newform 108.4.l.a.59.11

• Label: 108.4.l.a.59.11
• Level: $108$
• Weight: $4$
• Character: 108.59
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $312$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37220628062\)
Analytic rank:	\(0\)
Dimension:	\(312\)
Relative dimension:	\(52\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			59.11
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.10858 - 1.88517i) q^{2} +(-1.35776 + 5.01562i) q^{3} +(0.892234 + 7.95009i) q^{4} +(-5.18482 - 14.2452i) q^{5} +(12.3183 - 8.01624i) q^{6} +(-9.21752 + 1.62530i) q^{7} +(13.1060 - 18.4454i) q^{8} +(-23.3130 - 13.6200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(55\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.10858	−	1.88517i	−0.745496	−	0.666510i
							\(3\)	−1.35776	+	5.01562i	−0.261301	+	0.965257i
\(5\)	−5.18482	−	14.2452i	−0.463744	−	1.27413i								−0.922649	−	0.385642i	\(-0.873980\pi\)
							0.458904	−	0.888486i	\(-0.348242\pi\)
\(7\)	−9.21752	+	1.62530i	−0.497699	+	0.0877578i	−0.416863	−	0.908970i	\(-0.636870\pi\)
							−0.0808368	+	0.996727i	\(0.525759\pi\)
\(11\)	59.3614	+	21.6058i	1.62710	+	0.592217i	0.984716	−	0.174165i	\(-0.0557226\pi\)
							0.642385	+	0.766382i	\(0.277945\pi\)
\(13\)	44.8414	+	37.6264i	0.956675	+	0.802746i	0.980409	−	0.196973i	\(-0.0631110\pi\)
							−0.0237339	+	0.999718i	\(0.507555\pi\)
\(17\)	50.3759	+	29.0845i	0.718702	+	0.414943i	0.814275	−	0.580480i	\(-0.197135\pi\)
							−0.0955726	+	0.995422i	\(0.530468\pi\)
\(19\)	46.1670	−	26.6545i	0.557444	−	0.321840i	−0.194675	−	0.980868i	\(-0.562365\pi\)
							0.752119	+	0.659027i	\(0.229032\pi\)
\(23\)	2.42499	−	13.7528i	0.0219846	−	0.124681i	−0.971840	−	0.235640i	\(-0.924281\pi\)
							0.993825	+	0.110960i	\(0.0353924\pi\)
\(29\)	−12.2488	−	14.5975i	−0.0784326	−	0.0934723i	0.725401	−	0.688326i	\(-0.241654\pi\)
							−0.803834	+	0.594854i	\(0.797210\pi\)
\(31\)	197.673	+	34.8550i	1.14526	+	0.201940i	0.713905	−	0.700242i	\(-0.246925\pi\)
							0.431355	+	0.902182i	\(0.358036\pi\)
\(37\)	101.361	−	175.562i	0.450367	−	0.780058i	−0.548042	−	0.836451i	\(-0.684627\pi\)
							0.998409	+	0.0563927i	\(0.0179599\pi\)
\(41\)	197.308	−	235.142i	0.751567	−	0.895683i	−0.245716	−	0.969342i	\(-0.579023\pi\)
							0.997283	+	0.0736588i	\(0.0234676\pi\)
\(43\)	−182.426	+	501.213i	−0.646972	+	1.77754i	−0.0183458	+	0.999832i	\(0.505840\pi\)
							−0.628626	+	0.777708i	\(0.716382\pi\)
\(47\)	39.9568	+	226.606i	0.124006	+	0.703274i	0.981893	+	0.189436i	\(0.0606659\pi\)
							−0.857887	+	0.513839i	\(0.828223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.11	yes	312
4.3	odd	2	inner	108.4.l.a.59.18	yes	312
27.11	odd	18	inner	108.4.l.a.11.18	yes	312
108.11	even	18	inner	108.4.l.a.11.11	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.11	✓	312	108.11	even	18	inner
108.4.l.a.11.18	yes	312	27.11	odd	18	inner
108.4.l.a.59.11	yes	312	1.1	even	1	trivial
108.4.l.a.59.18	yes	312	4.3	odd	2	inner