Embedded newform 27.4.e.a.7.2

• Label: 27.4.e.a.7.2
• Level: $27$
• Weight: $4$
• Character: 27.7
• Analytic conductor: $1.593$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.59305157015\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	27.7
Dual form			27.4.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.05233 + 1.47493i) q^{2} +(4.37853 - 2.79794i) q^{3} +(8.11761 - 6.81148i) q^{4} +(3.22691 + 18.3007i) q^{5} +(-13.6165 + 17.7962i) q^{6} +(14.4087 + 12.0903i) q^{7} +(-5.59920 + 9.69809i) q^{8} +(11.3431 - 24.5017i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.05233	+	1.47493i	−1.43272	+	0.521466i	−0.937708	−	0.347424i	\(-0.887057\pi\)
							−0.495007	+	0.868889i	\(0.664834\pi\)
\(3\)	4.37853	−	2.79794i	0.842649	−	0.538464i
							\(5\)	3.22691	+	18.3007i	0.288624	+	1.63687i	0.692047	+	0.721853i	\(0.256709\pi\)
−0.403423	+	0.915013i	\(0.632180\pi\)
\(7\)	14.4087	+	12.0903i	0.777996	+	0.652816i	0.942743	−	0.333520i	\(-0.108236\pi\)
							−0.164747	+	0.986336i	\(0.552681\pi\)
\(11\)	−1.50172	+	8.51670i	−0.0411625	+	0.233444i	−0.998447	−	0.0557022i	\(-0.982260\pi\)
							0.957285	+	0.289146i	\(0.0933714\pi\)
\(13\)	−3.90473	−	1.42120i	−0.0833059	−	0.0303209i	0.300031	−	0.953929i	\(-0.403003\pi\)
							−0.383337	+	0.923609i	\(0.625225\pi\)
\(17\)	−59.2863	−	102.687i	−0.845826	−	1.46501i	−0.884902	−	0.465777i	\(-0.845775\pi\)
							0.0390759	−	0.999236i	\(-0.487559\pi\)
\(19\)	9.28989	−	16.0906i	0.112171	−	0.194286i	−0.804474	−	0.593987i	\(-0.797553\pi\)
							0.916645	+	0.399702i	\(0.130886\pi\)
\(23\)	−15.0361	+	12.6168i	−0.136315	+	0.114382i	−0.708396	−	0.705816i	\(-0.750581\pi\)
							0.572081	+	0.820197i	\(0.306136\pi\)
\(29\)	189.042	−	68.8055i	1.21049	−	0.440581i	0.343614	−	0.939111i	\(-0.388349\pi\)
							0.866873	+	0.498530i	\(0.166126\pi\)
\(31\)	−25.8306	+	21.6744i	−0.149655	+	0.125576i	−0.714541	−	0.699593i	\(-0.753365\pi\)
							0.564886	+	0.825169i	\(0.308920\pi\)
\(37\)	−26.0007	−	45.0345i	−0.115527	−	0.200098i	0.802463	−	0.596701i	\(-0.203522\pi\)
							−0.917990	+	0.396603i	\(0.870189\pi\)
\(41\)	89.8291	+	32.6951i	0.342169	+	0.124539i	0.507388	−	0.861717i	\(-0.330611\pi\)
							−0.165219	+	0.986257i	\(0.552833\pi\)
\(43\)	36.6219	−	207.693i	0.129879	−	0.736580i	−0.848411	−	0.529338i	\(-0.822440\pi\)
							0.978290	−	0.207242i	\(-0.0664486\pi\)
\(47\)	−276.865	−	232.318i	−0.859254	−	0.721000i	0.102553	−	0.994728i	\(-0.467299\pi\)
							−0.961807	+	0.273728i	\(0.911743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.7.2	yes	48
3.2	odd	2		81.4.e.a.19.7		48
9.2	odd	6		243.4.e.a.136.2		48
9.4	even	3		243.4.e.c.217.7		48
9.5	odd	6		243.4.e.b.217.2		48
9.7	even	3		243.4.e.d.136.7		48
27.2	odd	18		729.4.a.c.1.4		24
27.4	even	9	inner	27.4.e.a.4.2	✓	48
27.5	odd	18		243.4.e.b.28.2		48
27.13	even	9		243.4.e.d.109.7		48
27.14	odd	18		243.4.e.a.109.2		48
27.22	even	9		243.4.e.c.28.7		48
27.23	odd	18		81.4.e.a.64.7		48
27.25	even	9		729.4.a.d.1.21		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.4.2	✓	48	27.4	even	9	inner
27.4.e.a.7.2	yes	48	1.1	even	1	trivial
81.4.e.a.19.7		48	3.2	odd	2
81.4.e.a.64.7		48	27.23	odd	18
243.4.e.a.109.2		48	27.14	odd	18
243.4.e.a.136.2		48	9.2	odd	6
243.4.e.b.28.2		48	27.5	odd	18
243.4.e.b.217.2		48	9.5	odd	6
243.4.e.c.28.7		48	27.22	even	9
243.4.e.c.217.7		48	9.4	even	3
243.4.e.d.109.7		48	27.13	even	9
243.4.e.d.136.7		48	9.7	even	3
729.4.a.c.1.4		24	27.2	odd	18
729.4.a.d.1.21		24	27.25	even	9