Embedded newform 27.3.f.a.11.5

• Label: 27.3.f.a.11.5
• Level: $27$
• Weight: $3$
• Character: 27.11
• 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			11.5
Character	\(\chi\)	\(=\)	27.11
Dual form			27.3.f.a.5.5

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.5.5	✓	30	27.5	odd	18	inner
27.3.f.a.11.5	yes	30	1.1	even	1	trivial
81.3.f.a.35.1		30	3.2	odd	2
81.3.f.a.44.1		30	27.22	even	9
243.3.f.a.53.1		30	27.13	even	9
243.3.f.a.188.1		30	9.5	odd	6
243.3.f.b.26.5		30	9.2	odd	6
243.3.f.b.215.5		30	27.4	even	9
243.3.f.c.26.1		30	9.7	even	3
243.3.f.c.215.1		30	27.23	odd	18
243.3.f.d.53.5		30	27.14	odd	18
243.3.f.d.188.5		30	9.4	even	3
432.3.bc.a.65.4		30	4.3	odd	2
432.3.bc.a.113.4		30	108.59	even	18
729.3.b.a.728.4		30	27.7	even	9
729.3.b.a.728.27		30	27.20	odd	18