Embedded newform 28.6.f.a.19.18

• Label: 28.6.f.a.19.18
• 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			19.18
Character	\(\chi\)	\(=\)	28.19
Dual form			28.6.f.a.3.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.65684 + 0.0122071i) q^{2} +(11.9407 + 20.6819i) q^{3} +(31.9997 + 0.138107i) q^{4} +(-26.3592 - 15.2185i) q^{5} +(67.2942 + 117.140i) q^{6} +(-129.440 - 7.22687i) q^{7} +(181.016 + 1.17187i) q^{8} +(-163.661 + 283.469i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.65684	+	0.0122071i	0.999998	+	0.00215792i
							\(3\)	11.9407	+	20.6819i	0.765997	+	1.32674i	0.939718	+	0.341950i	\(0.111087\pi\)
−0.173722	+	0.984795i	\(0.555579\pi\)
\(5\)	−26.3592	−	15.2185i	−0.471528	−	0.272237i	0.245351	−	0.969434i	\(-0.421097\pi\)
							−0.716879	+	0.697198i	\(0.754430\pi\)
\(7\)	−129.440	−	7.22687i	−0.998445	−	0.0557449i
							\(11\)	633.622	−	365.822i	1.57888	−	0.911566i	0.583862	−	0.811853i	\(-0.301541\pi\)
0.995016	−	0.0997123i	\(-0.0317923\pi\)
\(13\)		−	555.117i		−	0.911016i	−0.890232	−	0.455508i	\(-0.849458\pi\)
							0.890232	−	0.455508i	\(-0.150542\pi\)
\(17\)	−1120.85	+	647.122i	−0.940643	+	0.543080i	−0.890162	−	0.455645i	\(-0.849409\pi\)
							−0.0504810	+	0.998725i	\(0.516075\pi\)
\(19\)	−387.571	+	671.292i	−0.246302	+	0.426607i	−0.962497	−	0.271293i	\(-0.912549\pi\)
							0.716195	+	0.697900i	\(0.245882\pi\)
\(23\)	−580.550	−	335.181i	−0.228834	−	0.132117i	0.381200	−	0.924492i	\(-0.375511\pi\)
							−0.610034	+	0.792375i	\(0.708844\pi\)
\(29\)	−3196.74			−0.705851			−0.352925	−	0.935652i	\(-0.614813\pi\)
							−0.352925	+	0.935652i	\(0.614813\pi\)
\(31\)	−156.048	−	270.284i	−0.0291645	−	0.0505144i	0.851075	−	0.525045i	\(-0.175951\pi\)
							−0.880239	+	0.474530i	\(0.842618\pi\)
\(37\)	−2553.96	+	4423.59i	−0.306697	+	0.531215i	−0.977638	−	0.210296i	\(-0.932557\pi\)
							0.670941	+	0.741511i	\(0.265891\pi\)
\(41\)			4987.32i			0.463349i	0.972793	+	0.231674i	\(0.0744204\pi\)
							−0.972793	+	0.231674i	\(0.925580\pi\)
\(43\)			12746.7i			1.05130i	0.850701	+	0.525650i	\(0.176178\pi\)
							−0.850701	+	0.525650i	\(0.823822\pi\)
\(47\)	−7560.18	+	13094.6i	−0.499215	+	0.864665i	−1.00000		0.000906592i	\(-0.999711\pi\)
							0.500785	+	0.865572i	\(0.333045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.6.f.a.19.18	yes	36
4.3	odd	2	inner	28.6.f.a.19.6	yes	36
7.2	even	3		196.6.d.b.195.13		36
7.3	odd	6	inner	28.6.f.a.3.6	✓	36
7.5	odd	6		196.6.d.b.195.14		36
28.3	even	6	inner	28.6.f.a.3.18	yes	36
28.19	even	6		196.6.d.b.195.15		36
28.23	odd	6		196.6.d.b.195.16		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.6	✓	36	7.3	odd	6	inner
28.6.f.a.3.18	yes	36	28.3	even	6	inner
28.6.f.a.19.6	yes	36	4.3	odd	2	inner
28.6.f.a.19.18	yes	36	1.1	even	1	trivial
196.6.d.b.195.13		36	7.2	even	3
196.6.d.b.195.14		36	7.5	odd	6
196.6.d.b.195.15		36	28.19	even	6
196.6.d.b.195.16		36	28.23	odd	6