Embedded newform 28.6.f.a.3.17

• Label: 28.6.f.a.3.17
• Level: $28$
• Weight: $6$
• Character: 28.3
• Analytic conductor: $4.491$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.49074695476\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3.17
Character	\(\chi\)	\(=\)	28.3
Dual form			28.6.f.a.19.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.63659 + 0.478421i) q^{2} +(-4.28235 + 7.41724i) q^{3} +(31.5422 + 5.39333i) q^{4} +(47.5979 - 27.4807i) q^{5} +(-27.6864 + 39.7592i) q^{6} +(6.17988 + 129.494i) q^{7} +(175.210 + 45.4904i) q^{8} +(84.8230 + 146.918i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.63659	+	0.478421i	0.996417	+	0.0845738i
							\(3\)	−4.28235	+	7.41724i	−0.274713	+	0.475816i	−0.970063	−	0.242855i	\(-0.921916\pi\)
0.695350	+	0.718671i	\(0.255249\pi\)
\(5\)	47.5979	−	27.4807i	0.851457	−	0.491589i	−0.00968526	−	0.999953i	\(-0.503083\pi\)
							0.861142	+	0.508364i	\(0.169750\pi\)
\(7\)	6.17988	+	129.494i	0.0476689	+	0.998863i
							\(11\)	−455.168	−	262.791i	−1.13420	−	0.654831i	−0.189212	−	0.981936i	\(-0.560593\pi\)
−0.944988	+	0.327105i	\(0.893927\pi\)
\(13\)		−	1005.77i		−	1.65060i	−0.564696	−	0.825299i	\(-0.691007\pi\)
							0.564696	−	0.825299i	\(-0.308993\pi\)
\(17\)	−1162.96	−	671.435i	−0.975983	−	0.563484i	−0.0749282	−	0.997189i	\(-0.523873\pi\)
							−0.901055	+	0.433705i	\(0.857206\pi\)
\(19\)	−23.5393	−	40.7713i	−0.0149592	−	0.0259102i	0.858449	−	0.512899i	\(-0.171429\pi\)
							−0.873408	+	0.486989i	\(0.838095\pi\)
\(23\)	−691.417	+	399.190i	−0.272534	+	0.157347i	−0.630039	−	0.776564i	\(-0.716961\pi\)
							0.357505	+	0.933911i	\(0.383628\pi\)
\(29\)	−600.350			−0.132559			−0.0662795	−	0.997801i	\(-0.521113\pi\)
							−0.0662795	+	0.997801i	\(0.521113\pi\)
\(31\)	2619.28	−	4536.73i	0.489529	−	0.847888i	−0.510399	−	0.859938i	\(-0.670502\pi\)
							0.999927	+	0.0120494i	\(0.00383555\pi\)
\(37\)	6609.88	+	11448.6i	0.793760	+	1.37483i	0.923624	+	0.383301i	\(0.125213\pi\)
							−0.129864	+	0.991532i	\(0.541454\pi\)
\(41\)			3464.72i			0.321891i	0.986963	+	0.160946i	\(0.0514543\pi\)
							−0.986963	+	0.160946i	\(0.948546\pi\)
\(43\)		−	11552.1i		−	0.952773i	−0.879236	−	0.476387i	\(-0.841946\pi\)
							0.879236	−	0.476387i	\(-0.158054\pi\)
\(47\)	1794.83	+	3108.74i	0.118517	+	0.205277i	0.919180	−	0.393838i	\(-0.128853\pi\)
							−0.800663	+	0.599114i	\(0.795519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.6.f.a.3.17	yes	36
4.3	odd	2	inner	28.6.f.a.3.7	✓	36
7.3	odd	6		196.6.d.b.195.11		36
7.4	even	3		196.6.d.b.195.12		36
7.5	odd	6	inner	28.6.f.a.19.7	yes	36
28.3	even	6		196.6.d.b.195.10		36
28.11	odd	6		196.6.d.b.195.9		36
28.19	even	6	inner	28.6.f.a.19.17	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.7	✓	36	4.3	odd	2	inner
28.6.f.a.3.17	yes	36	1.1	even	1	trivial
28.6.f.a.19.7	yes	36	7.5	odd	6	inner
28.6.f.a.19.17	yes	36	28.19	even	6	inner
196.6.d.b.195.9		36	28.11	odd	6
196.6.d.b.195.10		36	28.3	even	6
196.6.d.b.195.11		36	7.3	odd	6
196.6.d.b.195.12		36	7.4	even	3