Embedded newform 28.3.g.a.11.4

• Label: 28.3.g.a.11.4
• Level: $28$
• Weight: $3$
• Character: 28.11
• Analytic conductor: $0.763$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.762944740209\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 3*x^11 - 4*x^10 + 3*x^9 + 86*x^8 - 163*x^7 + 155*x^6 - 166*x^5 + 164*x^4 - 116*x^3 + 60*x^2 - 20*x + 4
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.4
Root			\(0.121721 + 0.507075i\) of defining polynomial
Character	\(\chi\)	\(=\)	28.11
Dual form			28.3.g.a.23.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.104798 + 1.99725i) q^{2} +(1.63031 + 0.941260i) q^{3} +(-3.97803 - 0.418616i) q^{4} +(-1.12649 - 1.95113i) q^{5} +(-2.05079 + 3.15750i) q^{6} +(6.84270 + 1.47562i) q^{7} +(1.25297 - 7.90127i) q^{8} +(-2.72806 - 4.72514i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 4 q^{4} - 2 q^{5} - 12 q^{6} - 8 q^{8} + 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.104798	+	1.99725i	−0.0523990	+	0.998626i
							\(3\)	1.63031	+	0.941260i	0.543437	+	0.313753i	0.746471	−	0.665418i	\(-0.231747\pi\)
−0.203034	+	0.979172i	\(0.565080\pi\)
\(5\)	−1.12649	−	1.95113i	−0.225297	−	0.390226i	0.731111	−	0.682258i	\(-0.239002\pi\)
							−0.956409	+	0.292032i	\(0.905669\pi\)
\(7\)	6.84270	+	1.47562i	0.977529	+	0.210803i
							\(11\)	−8.47301	−	4.89189i	−0.770274	−	0.444718i	0.0626986	−	0.998033i	\(-0.480029\pi\)
−0.832972	+	0.553315i	\(0.813363\pi\)
\(13\)	7.96206			0.612466			0.306233	−	0.951957i	\(-0.400931\pi\)
							0.306233	+	0.951957i	\(0.400931\pi\)
\(17\)	−13.1901	+	22.8460i	−0.775889	+	1.34388i	0.158404	+	0.987374i	\(0.449365\pi\)
							−0.934293	+	0.356505i	\(0.883968\pi\)
\(19\)	−21.1322	+	12.2007i	−1.11222	+	0.642140i	−0.939403	−	0.342814i	\(-0.888620\pi\)
							−0.172816	+	0.984954i	\(0.555287\pi\)
\(23\)	−3.55297	+	2.05131i	−0.154477	+	0.0891874i	−0.575246	−	0.817981i	\(-0.695094\pi\)
							0.420769	+	0.907168i	\(0.361760\pi\)
\(29\)	−12.3684			−0.426495			−0.213248	−	0.976998i	\(-0.568404\pi\)
							−0.213248	+	0.976998i	\(0.568404\pi\)
\(31\)	44.2877	+	25.5695i	1.42864	+	0.824823i	0.997013	−	0.0772309i	\(-0.0246079\pi\)
							0.431623	+	0.902054i	\(0.357941\pi\)
\(37\)	−16.7787	−	29.0615i	−0.453477	−	0.785446i	0.545122	−	0.838357i	\(-0.316483\pi\)
							−0.998599	+	0.0529109i	\(0.983150\pi\)
\(41\)	31.2806			0.762941			0.381471	−	0.924381i	\(-0.375418\pi\)
							0.381471	+	0.924381i	\(0.375418\pi\)
\(43\)			21.4052i			0.497796i	0.968530	+	0.248898i	\(0.0800685\pi\)
							−0.968530	+	0.248898i	\(0.919932\pi\)
\(47\)	39.4162	−	22.7569i	0.838642	−	0.484190i	−0.0181606	−	0.999835i	\(-0.505781\pi\)
							0.856802	+	0.515645i	\(0.172448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.3.g.a.11.4	✓	12
3.2	odd	2		252.3.y.c.235.3		12
4.3	odd	2	inner	28.3.g.a.11.6	yes	12
7.2	even	3	inner	28.3.g.a.23.6	yes	12
7.3	odd	6		196.3.c.h.99.1		6
7.4	even	3		196.3.c.i.99.1		6
7.5	odd	6		196.3.g.i.79.6		12
7.6	odd	2		196.3.g.i.67.4		12
8.3	odd	2		448.3.r.h.319.4		12
8.5	even	2		448.3.r.h.319.3		12
12.11	even	2		252.3.y.c.235.1		12
21.2	odd	6		252.3.y.c.163.1		12
28.3	even	6		196.3.c.h.99.2		6
28.11	odd	6		196.3.c.i.99.2		6
28.19	even	6		196.3.g.i.79.4		12
28.23	odd	6	inner	28.3.g.a.23.4	yes	12
28.27	even	2		196.3.g.i.67.6		12
56.37	even	6		448.3.r.h.191.4		12
56.51	odd	6		448.3.r.h.191.3		12
84.23	even	6		252.3.y.c.163.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.3.g.a.11.4	✓	12	1.1	even	1	trivial
28.3.g.a.11.6	yes	12	4.3	odd	2	inner
28.3.g.a.23.4	yes	12	28.23	odd	6	inner
28.3.g.a.23.6	yes	12	7.2	even	3	inner
196.3.c.h.99.1		6	7.3	odd	6
196.3.c.h.99.2		6	28.3	even	6
196.3.c.i.99.1		6	7.4	even	3
196.3.c.i.99.2		6	28.11	odd	6
196.3.g.i.67.4		12	7.6	odd	2
196.3.g.i.67.6		12	28.27	even	2
196.3.g.i.79.4		12	28.19	even	6
196.3.g.i.79.6		12	7.5	odd	6
252.3.y.c.163.1		12	21.2	odd	6
252.3.y.c.163.3		12	84.23	even	6
252.3.y.c.235.1		12	12.11	even	2
252.3.y.c.235.3		12	3.2	odd	2
448.3.r.h.191.3		12	56.51	odd	6
448.3.r.h.191.4		12	56.37	even	6
448.3.r.h.319.3		12	8.5	even	2
448.3.r.h.319.4		12	8.3	odd	2