Embedded newform 27.2.e.a.25.2

• Label: 27.2.e.a.25.2
• Level: $27$
• Weight: $2$
• Character: 27.25
• Analytic conductor: $0.216$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.215596085457\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	x^12 - 6*x^11 + 27*x^10 - 80*x^9 + 186*x^8 - 330*x^7 + 463*x^6 - 504*x^5 + 420*x^4 - 258*x^3 + 108*x^2 - 27*x + 3
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.2
Root			\(0.500000 - 1.00210i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.25
Dual form			27.2.e.a.13.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.183082 - 1.03831i) q^{2} +(-1.72962 + 0.0916693i) q^{3} +(0.834822 + 0.303850i) q^{4} +(-1.33735 + 1.12217i) q^{5} +(-0.221481 + 1.81266i) q^{6} +(-2.31094 + 0.841112i) q^{7} +(1.52266 - 2.63732i) q^{8} +(2.98319 - 0.317107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.183082	−	1.03831i	0.129458	−	0.734194i	−0.849101	−	0.528230i	\(-0.822856\pi\)
							0.978560	−	0.205964i	\(-0.0660330\pi\)
\(3\)	−1.72962	+	0.0916693i	−0.998598	+	0.0529253i
							\(5\)	−1.33735	+	1.12217i	−0.598081	+	0.501850i	−0.890828	−	0.454340i	\(-0.849875\pi\)
0.292747	+	0.956190i	\(0.405431\pi\)
\(7\)	−2.31094	+	0.841112i	−0.873452	+	0.317910i	−0.739564	−	0.673086i	\(-0.764968\pi\)
							−0.133888	+	0.990997i	\(0.542746\pi\)
\(11\)	−0.960783	−	0.806193i	−0.289687	−	0.243076i	0.486349	−	0.873764i	\(-0.338328\pi\)
							−0.776036	+	0.630688i	\(0.782773\pi\)
\(13\)	−0.789931	−	4.47992i	−0.219087	−	1.24251i	−0.873671	−	0.486517i	\(-0.838267\pi\)
							0.654584	−	0.755989i	\(-0.272844\pi\)
\(17\)	3.32358	+	5.75662i	0.806088	+	1.39618i	0.915554	+	0.402194i	\(0.131752\pi\)
							−0.109467	+	0.993990i	\(0.534914\pi\)
\(19\)	−0.124578	+	0.215776i	−0.0285802	+	0.0495023i	−0.879962	−	0.475045i	\(-0.842432\pi\)
							0.851382	+	0.524547i	\(0.175765\pi\)
\(23\)	−0.791222	−	0.287981i	−0.164981	−	0.0600483i	0.258209	−	0.966089i	\(-0.416868\pi\)
							−0.423190	+	0.906041i	\(0.639090\pi\)
\(29\)	−0.0889744	+	0.504599i	−0.0165221	+	0.0937016i	−0.991954	−	0.126601i	\(-0.959593\pi\)
							0.975432	+	0.220302i	\(0.0707044\pi\)
\(31\)	0.770551	+	0.280458i	0.138395	+	0.0503717i	0.410289	−	0.911956i	\(-0.365428\pi\)
							−0.271894	+	0.962327i	\(0.587650\pi\)
\(37\)	−1.30403	−	2.25865i	−0.214381	−	0.371319i	0.738700	−	0.674035i	\(-0.235440\pi\)
							−0.953081	+	0.302715i	\(0.902107\pi\)
\(41\)	−1.41572	−	8.02895i	−0.221099	−	1.25391i	−0.870005	−	0.493044i	\(-0.835884\pi\)
							0.648906	−	0.760869i	\(-0.275227\pi\)
\(43\)	3.31478	+	2.78143i	0.505500	+	0.424165i	0.859542	−	0.511065i	\(-0.170749\pi\)
							−0.354042	+	0.935229i	\(0.615193\pi\)
\(47\)	4.98256	−	1.81351i	0.726782	−	0.264527i	0.0479798	−	0.998848i	\(-0.484722\pi\)
							0.678802	+	0.734321i	\(0.262499\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.2.e.a.25.2	yes	12
3.2	odd	2		81.2.e.a.73.1		12
4.3	odd	2		432.2.u.c.241.2		12
5.2	odd	4		675.2.u.b.349.3		24
5.3	odd	4		675.2.u.b.349.2		24
5.4	even	2		675.2.l.c.376.1		12
9.2	odd	6		243.2.e.a.55.2		12
9.4	even	3		243.2.e.c.136.2		12
9.5	odd	6		243.2.e.b.136.1		12
9.7	even	3		243.2.e.d.55.1		12
27.2	odd	18		729.2.c.b.487.5		12
27.4	even	9		243.2.e.c.109.2		12
27.5	odd	18		243.2.e.a.190.2		12
27.7	even	9		729.2.c.e.244.2		12
27.11	odd	18		729.2.a.d.1.2		6
27.13	even	9	inner	27.2.e.a.13.2	✓	12
27.14	odd	18		81.2.e.a.10.1		12
27.16	even	9		729.2.a.a.1.5		6
27.20	odd	18		729.2.c.b.244.5		12
27.22	even	9		243.2.e.d.190.1		12
27.23	odd	18		243.2.e.b.109.1		12
27.25	even	9		729.2.c.e.487.2		12
108.67	odd	18		432.2.u.c.337.2		12
135.13	odd	36		675.2.u.b.499.3		24
135.67	odd	36		675.2.u.b.499.2		24
135.94	even	18		675.2.l.c.526.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.2.e.a.13.2	✓	12	27.13	even	9	inner
27.2.e.a.25.2	yes	12	1.1	even	1	trivial
81.2.e.a.10.1		12	27.14	odd	18
81.2.e.a.73.1		12	3.2	odd	2
243.2.e.a.55.2		12	9.2	odd	6
243.2.e.a.190.2		12	27.5	odd	18
243.2.e.b.109.1		12	27.23	odd	18
243.2.e.b.136.1		12	9.5	odd	6
243.2.e.c.109.2		12	27.4	even	9
243.2.e.c.136.2		12	9.4	even	3
243.2.e.d.55.1		12	9.7	even	3
243.2.e.d.190.1		12	27.22	even	9
432.2.u.c.241.2		12	4.3	odd	2
432.2.u.c.337.2		12	108.67	odd	18
675.2.l.c.376.1		12	5.4	even	2
675.2.l.c.526.1		12	135.94	even	18
675.2.u.b.349.2		24	5.3	odd	4
675.2.u.b.349.3		24	5.2	odd	4
675.2.u.b.499.2		24	135.67	odd	36
675.2.u.b.499.3		24	135.13	odd	36
729.2.a.a.1.5		6	27.16	even	9
729.2.a.d.1.2		6	27.11	odd	18
729.2.c.b.244.5		12	27.20	odd	18
729.2.c.b.487.5		12	27.2	odd	18
729.2.c.e.244.2		12	27.7	even	9
729.2.c.e.487.2		12	27.25	even	9