Embedded newform 28.6.f.a.3.12

• Label: 28.6.f.a.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	28.3
Dual form			28.6.f.a.19.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.36282 + 5.13976i) q^{2} +(-9.72416 + 16.8427i) q^{3} +(-20.8342 + 24.2886i) q^{4} +(43.6962 - 25.2280i) q^{5} +(-109.544 - 10.1835i) q^{6} +(-129.558 - 4.64978i) q^{7} +(-174.065 - 49.6932i) q^{8} +(-67.6187 - 117.119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.36282	+	5.13976i	0.417691	+	0.908589i
							\(3\)	−9.72416	+	16.8427i	−0.623805	+	1.08046i	0.364965	+	0.931021i	\(0.381081\pi\)
−0.988771	+	0.149441i	\(0.952252\pi\)
\(5\)	43.6962	−	25.2280i	0.781662	−	0.451293i	−0.0553570	−	0.998467i	\(-0.517630\pi\)
							0.837019	+	0.547174i	\(0.184296\pi\)
\(7\)	−129.558	−	4.64978i	−0.999357	−	0.0358664i
							\(11\)	244.393	+	141.100i	0.608986	+	0.351598i	0.772569	−	0.634931i	\(-0.218972\pi\)
−0.163582	+	0.986530i	\(0.552305\pi\)
\(13\)			783.423i			1.28570i	0.765994	+	0.642848i	\(0.222247\pi\)
							−0.765994	+	0.642848i	\(0.777753\pi\)
\(17\)	−397.687	−	229.604i	−0.333748	−	0.192689i	0.323756	−	0.946141i	\(-0.395054\pi\)
							−0.657504	+	0.753451i	\(0.728388\pi\)
\(19\)	1314.37	+	2276.56i	0.835283	+	1.44675i	0.893800	+	0.448466i	\(0.148030\pi\)
							−0.0585167	+	0.998286i	\(0.518637\pi\)
\(23\)	2674.39	−	1544.06i	1.05416	−	0.608618i	0.130346	−	0.991469i	\(-0.458391\pi\)
							0.923810	+	0.382851i	\(0.125058\pi\)
\(29\)	2371.45			0.523623			0.261811	−	0.965119i	\(-0.415680\pi\)
							0.261811	+	0.965119i	\(0.415680\pi\)
\(31\)	938.208	−	1625.02i	0.175346	−	0.303708i	−0.764935	−	0.644107i	\(-0.777229\pi\)
							0.940281	+	0.340400i	\(0.110562\pi\)
\(37\)	−1795.63	−	3110.11i	−0.215631	−	0.373484i	0.737837	−	0.674979i	\(-0.235848\pi\)
							−0.953468	+	0.301495i	\(0.902514\pi\)
\(41\)			12855.3i			1.19432i	0.802121	+	0.597162i	\(0.203705\pi\)
							−0.802121	+	0.597162i	\(0.796295\pi\)
\(43\)		−	20317.1i		−	1.67568i	−0.545915	−	0.837841i	\(-0.683818\pi\)
							0.545915	−	0.837841i	\(-0.316182\pi\)
\(47\)	5317.85	+	9210.78i	0.351149	+	0.608208i	0.986451	−	0.164056i	\(-0.0524578\pi\)
							−0.635302	+	0.772264i	\(0.719124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.6.f.a.3.12	✓	36
4.3	odd	2	inner	28.6.f.a.3.13	yes	36
7.3	odd	6		196.6.d.b.195.1		36
7.4	even	3		196.6.d.b.195.2		36
7.5	odd	6	inner	28.6.f.a.19.13	yes	36
28.3	even	6		196.6.d.b.195.4		36
28.11	odd	6		196.6.d.b.195.3		36
28.19	even	6	inner	28.6.f.a.19.12	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.12	✓	36	1.1	even	1	trivial
28.6.f.a.3.13	yes	36	4.3	odd	2	inner
28.6.f.a.19.12	yes	36	28.19	even	6	inner
28.6.f.a.19.13	yes	36	7.5	odd	6	inner
196.6.d.b.195.1		36	7.3	odd	6
196.6.d.b.195.2		36	7.4	even	3
196.6.d.b.195.3		36	28.11	odd	6
196.6.d.b.195.4		36	28.3	even	6