Embedded newform 28.6.f.a.19.12

• Label: 28.6.f.a.19.12
• Level: $28$
• Weight: $6$
• Character: 28.19
• Analytic conductor: $4.491$
• Analytic rank: $0$
• Dimension: $36$
• 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			19.12
Character	\(\chi\)	\(=\)	28.19
Dual form			28.6.f.a.3.12

$q$-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 \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.12	✓	36	28.3	even	6	inner
28.6.f.a.3.13	yes	36	7.3	odd	6	inner
28.6.f.a.19.12	yes	36	1.1	even	1	trivial
28.6.f.a.19.13	yes	36	4.3	odd	2	inner
196.6.d.b.195.1		36	28.23	odd	6
196.6.d.b.195.2		36	28.19	even	6
196.6.d.b.195.3		36	7.5	odd	6
196.6.d.b.195.4		36	7.2	even	3