Embedded newform 108.4.l.a.59.14

• Label: 108.4.l.a.59.14
• 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			59.14
Character	\(\chi\)	\(=\)	108.59
Dual form			108.4.l.a.11.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88673 - 2.10719i) q^{2} +(2.88986 + 4.31841i) q^{3} +(-0.880530 + 7.95139i) q^{4} +(3.28651 + 9.02962i) q^{5} +(3.64735 - 14.2372i) q^{6} +(-18.0956 + 3.19074i) q^{7} +(18.4164 - 13.1467i) q^{8} +(-10.2974 + 24.9592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 312 q - 6 q^{2} - 6 q^{4} - 12 q^{5} - 6 q^{6} - 9 q^{8} - 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88673	−	2.10719i	−0.667058	−	0.745005i
							\(3\)	2.88986	+	4.31841i	0.556154	+	0.831079i
\(5\)	3.28651	+	9.02962i	0.293955	+	0.807634i								0.995478	+	0.0949876i	\(0.0302812\pi\)
							−0.701524	+	0.712646i	\(0.747497\pi\)
\(7\)	−18.0956	+	3.19074i	−0.977070	+	0.172284i	−0.639311	−	0.768949i	\(-0.720780\pi\)
							−0.337760	+	0.941232i	\(0.609669\pi\)
\(11\)	−49.0304	−	17.8456i	−1.34393	−	0.489150i	−0.432882	−	0.901451i	\(-0.642503\pi\)
							−0.911048	+	0.412300i	\(0.864725\pi\)
\(13\)	−45.2507	−	37.9699i	−0.965407	−	0.810073i	0.0164169	−	0.999865i	\(-0.494774\pi\)
							−0.981824	+	0.189792i	\(0.939219\pi\)
\(17\)	48.9268	+	28.2479i	0.698028	+	0.403007i	0.806613	−	0.591080i	\(-0.201298\pi\)
							−0.108584	+	0.994087i	\(0.534632\pi\)
\(19\)	11.8846	−	6.86160i	0.143501	−	0.0828505i	−0.426530	−	0.904473i	\(-0.640264\pi\)
							0.570032	+	0.821623i	\(0.306931\pi\)
\(23\)	−32.1761	+	182.479i	−0.291703	+	1.65433i	0.388605	+	0.921404i	\(0.372957\pi\)
							−0.680308	+	0.732926i	\(0.738154\pi\)
\(29\)	199.157	+	237.347i	1.27526	+	1.51980i	0.734467	+	0.678645i	\(0.237432\pi\)
							0.540796	+	0.841154i	\(0.318123\pi\)
\(31\)	−123.956	−	21.8568i	−0.718167	−	0.126632i	−0.197390	−	0.980325i	\(-0.563247\pi\)
							−0.520777	+	0.853693i	\(0.674358\pi\)
\(37\)	41.7796	−	72.3645i	0.185636	−	0.321531i	−0.758155	−	0.652075i	\(-0.773899\pi\)
							0.943791	+	0.330544i	\(0.107232\pi\)
\(41\)	29.6614	−	35.3491i	0.112984	−	0.134649i	−0.706588	−	0.707625i	\(-0.749767\pi\)
							0.819572	+	0.572976i	\(0.194211\pi\)
\(43\)	−117.701	+	323.380i	−0.417422	+	1.14686i	0.535736	+	0.844386i	\(0.320034\pi\)
							−0.953158	+	0.302473i	\(0.902188\pi\)
\(47\)	−40.4260	−	229.267i	−0.125463	−	0.711533i	−0.981032	−	0.193846i	\(-0.937904\pi\)
							0.855570	−	0.517688i	\(-0.173207\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.4.l.a.59.14	yes	312
4.3	odd	2	inner	108.4.l.a.59.17	yes	312
27.11	odd	18	inner	108.4.l.a.11.17	yes	312
108.11	even	18	inner	108.4.l.a.11.14	✓	312

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.l.a.11.14	✓	312	108.11	even	18	inner
108.4.l.a.11.17	yes	312	27.11	odd	18	inner
108.4.l.a.59.14	yes	312	1.1	even	1	trivial
108.4.l.a.59.17	yes	312	4.3	odd	2	inner