Embedded newform 27.3.f.a.5.5

• Label: 27.3.f.a.5.5
• Level: $27$
• Weight: $3$
• Character: 27.5
• 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			5.5
Character	\(\chi\)	\(=\)	27.5
Dual form			27.3.f.a.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.13670 + 2.54642i) q^{2} +(-2.03568 - 2.20363i) q^{3} +(-1.22417 + 6.94260i) q^{4} +(-2.35247 - 6.46335i) q^{5} +(1.26173 - 9.89219i) q^{6} +(1.10811 + 6.28443i) q^{7} +(-8.77937 + 5.06877i) q^{8} +(-0.712002 + 8.97179i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 15 q^{5} - 18 q^{6} - 6 q^{7} - 9 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.13670	+	2.54642i	1.06835	+	1.27321i	0.960272	+	0.279065i	\(0.0900245\pi\)
							0.108076	+	0.994143i	\(0.465531\pi\)
\(3\)	−2.03568	−	2.20363i	−0.678560	−	0.734545i
							\(5\)	−2.35247	−	6.46335i	−0.470493	−	1.29267i	−0.917356	−	0.398067i	\(-0.869681\pi\)
0.446863	−	0.894602i	\(-0.352541\pi\)
\(7\)	1.10811	+	6.28443i	0.158302	+	0.897775i	0.955705	+	0.294328i	\(0.0950957\pi\)
							−0.797403	+	0.603448i	\(0.793793\pi\)
\(11\)	0.425636	−	1.16942i	0.0386941	−	0.106311i	−0.918841	−	0.394628i	\(-0.870873\pi\)
							0.957535	+	0.288317i	\(0.0930956\pi\)
\(13\)	−4.09039	−	3.43225i	−0.314646	−	0.264019i	0.471763	−	0.881725i	\(-0.343618\pi\)
							−0.786409	+	0.617706i	\(0.788062\pi\)
\(17\)	−13.7418	−	7.93385i	−0.808343	−	0.466697i	0.0380373	−	0.999276i	\(-0.487889\pi\)
							−0.846380	+	0.532579i	\(0.821223\pi\)
\(19\)	6.78917	+	11.7592i	0.357325	+	0.618905i	0.987513	−	0.157538i	\(-0.0503556\pi\)
							−0.630188	+	0.776442i	\(0.717022\pi\)
\(23\)	24.4347	+	4.30850i	1.06238	+	0.187326i	0.677412	−	0.735604i	\(-0.263101\pi\)
							0.384968	+	0.922930i	\(0.374213\pi\)
\(29\)	−19.9186	−	23.7381i	−0.686849	−	0.818554i	0.304122	−	0.952633i	\(-0.401637\pi\)
							−0.990971	+	0.134079i	\(0.957192\pi\)
\(31\)	−2.75731	+	15.6375i	−0.0889455	+	0.504435i	0.907490	+	0.420074i	\(0.137996\pi\)
							−0.996435	+	0.0843610i	\(0.973115\pi\)
\(37\)	26.0365	−	45.0965i	0.703688	−	1.21882i	−0.263474	−	0.964666i	\(-0.584868\pi\)
							0.967163	−	0.254158i	\(-0.0817982\pi\)
\(41\)	−0.694475	+	0.827643i	−0.0169384	+	0.0201864i	−0.774447	−	0.632639i	\(-0.781972\pi\)
							0.757509	+	0.652825i	\(0.226416\pi\)
\(43\)	−45.3904	−	16.5207i	−1.05559	−	0.384203i	−0.244820	−	0.969569i	\(-0.578729\pi\)
							−0.810770	+	0.585365i	\(0.800951\pi\)
\(47\)	−56.7291	+	10.0029i	−1.20700	+	0.212827i	−0.740723	−	0.671811i	\(-0.765517\pi\)
							−0.466279	+	0.884638i	\(0.654406\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.3.f.a.5.5	✓	30
3.2	odd	2		81.3.f.a.44.1		30
4.3	odd	2		432.3.bc.a.113.4		30
9.2	odd	6		243.3.f.a.53.1		30
9.4	even	3		243.3.f.c.215.1		30
9.5	odd	6		243.3.f.b.215.5		30
9.7	even	3		243.3.f.d.53.5		30
27.2	odd	18		243.3.f.d.188.5		30
27.4	even	9		729.3.b.a.728.27		30
27.7	even	9		243.3.f.b.26.5		30
27.11	odd	18	inner	27.3.f.a.11.5	yes	30
27.16	even	9		81.3.f.a.35.1		30
27.20	odd	18		243.3.f.c.26.1		30
27.23	odd	18		729.3.b.a.728.4		30
27.25	even	9		243.3.f.a.188.1		30
108.11	even	18		432.3.bc.a.65.4		30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.5.5	✓	30	1.1	even	1	trivial
27.3.f.a.11.5	yes	30	27.11	odd	18	inner
81.3.f.a.35.1		30	27.16	even	9
81.3.f.a.44.1		30	3.2	odd	2
243.3.f.a.53.1		30	9.2	odd	6
243.3.f.a.188.1		30	27.25	even	9
243.3.f.b.26.5		30	27.7	even	9
243.3.f.b.215.5		30	9.5	odd	6
243.3.f.c.26.1		30	27.20	odd	18
243.3.f.c.215.1		30	9.4	even	3
243.3.f.d.53.5		30	9.7	even	3
243.3.f.d.188.5		30	27.2	odd	18
432.3.bc.a.65.4		30	108.11	even	18
432.3.bc.a.113.4		30	4.3	odd	2
729.3.b.a.728.4		30	27.23	odd	18
729.3.b.a.728.27		30	27.4	even	9