Embedded newform 28.4.f.a.3.7

• Label: 28.4.f.a.3.7
• Level: $28$
• Weight: $4$
• Character: 28.3
• Analytic conductor: $1.652$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	28.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.65205348016\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 2*x^18 - 24*x^17 + 28*x^16 + 56*x^15 - 192*x^14 + 352*x^13 - 448*x^12 + 5376*x^11 - 41472*x^10 + 43008*x^9 - 28672*x^8 + 180224*x^7 - 786432*x^6 + 1835008*x^5 + 7340032*x^4 - 50331648*x^3 - 33554432*x^2 + 1073741824
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{24} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			3.7
Root			\(-2.82600 + 0.117237i\) of defining polynomial
Character	\(\chi\)	\(=\)	28.3
Dual form			28.4.f.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.31147 - 2.50600i) q^{2} +(-0.0469307 + 0.0812864i) q^{3} +(-4.56010 - 6.57309i) q^{4} +(12.4861 - 7.20883i) q^{5} +(0.142156 + 0.224213i) q^{6} +(-15.0686 + 10.7674i) q^{7} +(-22.4526 + 2.80725i) q^{8} +(13.4956 + 23.3751i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{4} - 6 q^{5} + 72 q^{8} - 56 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.31147	−	2.50600i	0.463674	−	0.886006i
							\(3\)	−0.0469307	+	0.0812864i	−0.00903182	+	0.0156436i	−0.870506	−	0.492158i	\(-0.836208\pi\)
0.861474	+	0.507801i	\(0.169542\pi\)
\(5\)	12.4861	−	7.20883i	1.11679	−	0.644778i	0.176209	−	0.984353i	\(-0.443617\pi\)
							0.940579	+	0.339575i	\(0.110283\pi\)
\(7\)	−15.0686	+	10.7674i	−0.813627	+	0.581387i
							\(11\)	31.6215	+	18.2567i	0.866749	+	0.500418i	0.866267	−	0.499582i	\(-0.166513\pi\)
0.000482758		1.00000i	\(0.499846\pi\)
\(13\)			15.3232i			0.326914i	0.986550	+	0.163457i	\(0.0522645\pi\)
							−0.986550	+	0.163457i	\(0.947735\pi\)
\(17\)	−54.1508	−	31.2640i	−0.772558	−	0.446037i	0.0612284	−	0.998124i	\(-0.480498\pi\)
							−0.833786	+	0.552087i	\(0.813832\pi\)
\(19\)	−34.6779	−	60.0639i	−0.418719	−	0.725242i	0.577092	−	0.816679i	\(-0.304187\pi\)
							−0.995811	+	0.0914368i	\(0.970854\pi\)
\(23\)	−12.3839	+	7.14987i	−0.112271	+	0.0648196i	−0.555084	−	0.831794i	\(-0.687314\pi\)
							0.442813	+	0.896614i	\(0.353981\pi\)
\(29\)	−157.182			−1.00648			−0.503241	−	0.864146i	\(-0.667859\pi\)
							−0.503241	+	0.864146i	\(0.667859\pi\)
\(31\)	165.570	−	286.776i	0.959266	−	1.66150i	0.234977	−	0.972001i	\(-0.424499\pi\)
							0.724289	−	0.689496i	\(-0.242168\pi\)
\(37\)	−38.0629	−	65.9269i	−0.169122	−	0.292927i	0.768990	−	0.639261i	\(-0.220760\pi\)
							−0.938111	+	0.346334i	\(0.887426\pi\)
\(41\)			335.509i			1.27799i	0.769209	+	0.638997i	\(0.220650\pi\)
							−0.769209	+	0.638997i	\(0.779350\pi\)
\(43\)			484.545i			1.71843i	0.511617	+	0.859213i	\(0.329047\pi\)
							−0.511617	+	0.859213i	\(0.670953\pi\)
\(47\)	−187.406	−	324.597i	−0.581617	−	1.00739i	−0.995288	−	0.0969640i	\(-0.969087\pi\)
							0.413671	−	0.910427i	\(-0.364247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.4.f.a.3.7	yes	20
4.3	odd	2	inner	28.4.f.a.3.1	✓	20
7.2	even	3		196.4.f.d.19.1		20
7.3	odd	6		196.4.d.b.195.15		20
7.4	even	3		196.4.d.b.195.16		20
7.5	odd	6	inner	28.4.f.a.19.1	yes	20
7.6	odd	2		196.4.f.d.31.7		20
8.3	odd	2		448.4.p.h.255.5		20
8.5	even	2		448.4.p.h.255.6		20
28.3	even	6		196.4.d.b.195.14		20
28.11	odd	6		196.4.d.b.195.13		20
28.19	even	6	inner	28.4.f.a.19.7	yes	20
28.23	odd	6		196.4.f.d.19.7		20
28.27	even	2		196.4.f.d.31.1		20
56.5	odd	6		448.4.p.h.383.5		20
56.19	even	6		448.4.p.h.383.6		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.f.a.3.1	✓	20	4.3	odd	2	inner
28.4.f.a.3.7	yes	20	1.1	even	1	trivial
28.4.f.a.19.1	yes	20	7.5	odd	6	inner
28.4.f.a.19.7	yes	20	28.19	even	6	inner
196.4.d.b.195.13		20	28.11	odd	6
196.4.d.b.195.14		20	28.3	even	6
196.4.d.b.195.15		20	7.3	odd	6
196.4.d.b.195.16		20	7.4	even	3
196.4.f.d.19.1		20	7.2	even	3
196.4.f.d.19.7		20	28.23	odd	6
196.4.f.d.31.1		20	28.27	even	2
196.4.f.d.31.7		20	7.6	odd	2
448.4.p.h.255.5		20	8.3	odd	2
448.4.p.h.255.6		20	8.5	even	2
448.4.p.h.383.5		20	56.5	odd	6
448.4.p.h.383.6		20	56.19	even	6