Embedded newform 27.3.f.a.2.3

• Label: 27.3.f.a.2.3
• Level: $27$
• Weight: $3$
• Character: 27.2
• Analytic conductor: $0.736$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.735696713773\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.3
Character	\(\chi\)	\(=\)	27.2
Dual form			27.3.f.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.115908 + 0.0204377i) q^{2} +(2.32380 - 1.89736i) q^{3} +(-3.74575 - 1.36334i) q^{4} +(3.98394 + 4.74788i) q^{5} +(0.308125 - 0.172426i) q^{6} +(-7.49258 + 2.72708i) q^{7} +(-0.814011 - 0.469969i) q^{8} +(1.80008 - 8.81815i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.115908	+	0.0204377i	0.0579540	+	0.0102189i	0.202550	−	0.979272i	\(-0.435077\pi\)
							−0.144596	+	0.989491i	\(0.546188\pi\)
\(3\)	2.32380	−	1.89736i	0.774600	−	0.632452i
							\(5\)	3.98394	+	4.74788i	0.796788	+	0.949575i	0.999560	−	0.0296506i	\(-0.00943948\pi\)
−0.202772	+	0.979226i	\(0.564995\pi\)
\(7\)	−7.49258	+	2.72708i	−1.07037	+	0.389582i	−0.816315	−	0.577608i	\(-0.803986\pi\)
							−0.254054	+	0.967190i	\(0.581764\pi\)
\(11\)	−7.35571	+	8.76619i	−0.668701	+	0.796927i	−0.988607	−	0.150523i	\(-0.951904\pi\)
							0.319906	+	0.947449i	\(0.396349\pi\)
\(13\)	−1.43358	−	8.13025i	−0.110276	−	0.625403i	−0.988981	−	0.148040i	\(-0.952704\pi\)
							0.878706	−	0.477364i	\(-0.158408\pi\)
\(17\)	20.7128	−	11.9585i	1.21840	−	0.703443i	0.253824	−	0.967250i	\(-0.418311\pi\)
							0.964576	+	0.263807i	\(0.0849782\pi\)
\(19\)	6.12102	−	10.6019i	0.322159	−	0.557995i	−0.658774	−	0.752341i	\(-0.728925\pi\)
							0.980933	+	0.194345i	\(0.0622582\pi\)
\(23\)	−5.04566	+	13.8628i	−0.219377	+	0.602732i	−0.999745	−	0.0225880i	\(-0.992809\pi\)
							0.780368	+	0.625320i	\(0.215032\pi\)
\(29\)	−17.5228	−	3.08975i	−0.604235	−	0.106543i	−0.136843	−	0.990593i	\(-0.543695\pi\)
							−0.467393	+	0.884050i	\(0.654807\pi\)
\(31\)	−12.2356	−	4.45339i	−0.394697	−	0.143658i	0.137044	−	0.990565i	\(-0.456240\pi\)
							−0.531741	+	0.846907i	\(0.678462\pi\)
\(37\)	8.53500	+	14.7831i	0.230676	+	0.399542i	0.958007	−	0.286744i	\(-0.0925730\pi\)
							−0.727331	+	0.686286i	\(0.759240\pi\)
\(41\)	30.2688	−	5.33721i	0.738265	−	0.130176i	0.208144	−	0.978098i	\(-0.433258\pi\)
							0.530121	+	0.847922i	\(0.322147\pi\)
\(43\)	−30.5116	−	25.6022i	−0.709571	−	0.595401i	0.214908	−	0.976634i	\(-0.431055\pi\)
							−0.924479	+	0.381233i	\(0.875499\pi\)
\(47\)	19.9668	+	54.8583i	0.424825	+	1.16720i	0.948914	+	0.315534i	\(0.102184\pi\)
							−0.524089	+	0.851663i	\(0.675594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.2.3	✓	30
3.2	odd	2		81.3.f.a.8.3		30
4.3	odd	2		432.3.bc.a.353.1		30
9.2	odd	6		243.3.f.b.188.3		30
9.4	even	3		243.3.f.d.107.3		30
9.5	odd	6		243.3.f.a.107.3		30
9.7	even	3		243.3.f.c.188.3		30
27.4	even	9		243.3.f.a.134.3		30
27.5	odd	18		243.3.f.c.53.3		30
27.11	odd	18		729.3.b.a.728.15		30
27.13	even	9		81.3.f.a.71.3		30
27.14	odd	18	inner	27.3.f.a.14.3	yes	30
27.16	even	9		729.3.b.a.728.16		30
27.22	even	9		243.3.f.b.53.3		30
27.23	odd	18		243.3.f.d.134.3		30
108.95	even	18		432.3.bc.a.257.1		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.3	✓	30	1.1	even	1	trivial
27.3.f.a.14.3	yes	30	27.14	odd	18	inner
81.3.f.a.8.3		30	3.2	odd	2
81.3.f.a.71.3		30	27.13	even	9
243.3.f.a.107.3		30	9.5	odd	6
243.3.f.a.134.3		30	27.4	even	9
243.3.f.b.53.3		30	27.22	even	9
243.3.f.b.188.3		30	9.2	odd	6
243.3.f.c.53.3		30	27.5	odd	18
243.3.f.c.188.3		30	9.7	even	3
243.3.f.d.107.3		30	9.4	even	3
243.3.f.d.134.3		30	27.23	odd	18
432.3.bc.a.257.1		30	108.95	even	18
432.3.bc.a.353.1		30	4.3	odd	2
729.3.b.a.728.15		30	27.11	odd	18
729.3.b.a.728.16		30	27.16	even	9