Embedded newform 28.5.g.a.11.6

• Label: 28.5.g.a.11.6
• Level: $28$
• Weight: $5$
• Character: 28.11
• Analytic conductor: $2.894$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.89435896635\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	28.11
Dual form			28.5.g.a.23.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23058 - 3.80601i) q^{2} +(-8.26641 - 4.77262i) q^{3} +(-12.9714 + 9.36716i) q^{4} +(8.72165 + 15.1063i) q^{5} +(-7.99215 + 37.3351i) q^{6} +(-48.2343 + 8.62858i) q^{7} +(51.6137 + 37.8421i) q^{8} +(5.05573 + 8.75678i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{2} - 4 q^{4} - 2 q^{5} - 36 q^{6} - 184 q^{8} + 268 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23058	−	3.80601i	−0.307644	−	0.951501i
							\(3\)	−8.26641	−	4.77262i	−0.918490	−	0.530291i	−0.0353372	−	0.999375i	\(-0.511251\pi\)
−0.883153	+	0.469085i	\(0.844584\pi\)
\(5\)	8.72165	+	15.1063i	0.348866	+	0.604254i	0.986048	−	0.166459i	\(-0.0532334\pi\)
							−0.637182	+	0.770713i	\(0.719900\pi\)
\(7\)	−48.2343	+	8.62858i	−0.984373	+	0.176093i
							\(11\)	−118.403	−	68.3603i	−0.978541	−	0.564961i	−0.0767119	−	0.997053i	\(-0.524442\pi\)
−0.901829	+	0.432092i	\(0.857775\pi\)
\(13\)	−190.457			−1.12697			−0.563483	−	0.826128i	\(-0.690539\pi\)
							−0.563483	+	0.826128i	\(0.690539\pi\)
\(17\)	39.8678	−	69.0530i	0.137951	−	0.238938i	−0.788770	−	0.614688i	\(-0.789282\pi\)
							0.926721	+	0.375751i	\(0.122615\pi\)
\(19\)	377.648	−	218.035i	1.04612	−	0.603976i	0.124557	−	0.992212i	\(-0.460249\pi\)
							0.921560	+	0.388237i	\(0.126916\pi\)
\(23\)	−481.324	+	277.892i	−0.909874	+	0.525316i	−0.880391	−	0.474249i	\(-0.842720\pi\)
							−0.0294836	+	0.999565i	\(0.509386\pi\)
\(29\)	−1033.21			−1.22855			−0.614275	−	0.789092i	\(-0.710551\pi\)
							−0.614275	+	0.789092i	\(0.710551\pi\)
\(31\)	888.938	+	513.229i	0.925014	+	0.534057i	0.885231	−	0.465152i	\(-0.154000\pi\)
							0.0397825	+	0.999208i	\(0.487334\pi\)
\(37\)	−922.413	−	1597.67i	−0.673786	−	1.16703i	−0.976822	−	0.214052i	\(-0.931334\pi\)
							0.303036	−	0.952979i	\(-0.402000\pi\)
\(41\)	1438.20			0.855563			0.427782	−	0.903882i	\(-0.359295\pi\)
							0.427782	+	0.903882i	\(0.359295\pi\)
\(43\)		−	356.032i		−	0.192554i	−0.995355	−	0.0962768i	\(-0.969307\pi\)
							0.995355	−	0.0962768i	\(-0.0306934\pi\)
\(47\)	436.955	−	252.276i	0.197807	−	0.114204i	−0.397825	−	0.917461i	\(-0.630235\pi\)
							0.595632	+	0.803257i	\(0.296902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.5.g.a.11.6	yes	28
4.3	odd	2	inner	28.5.g.a.11.4	✓	28
7.2	even	3	inner	28.5.g.a.23.4	yes	28
7.3	odd	6		196.5.c.g.99.14		14
7.4	even	3		196.5.c.h.99.14		14
28.3	even	6		196.5.c.g.99.13		14
28.11	odd	6		196.5.c.h.99.13		14
28.23	odd	6	inner	28.5.g.a.23.6	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.g.a.11.4	✓	28	4.3	odd	2	inner
28.5.g.a.11.6	yes	28	1.1	even	1	trivial
28.5.g.a.23.4	yes	28	7.2	even	3	inner
28.5.g.a.23.6	yes	28	28.23	odd	6	inner
196.5.c.g.99.13		14	28.3	even	6
196.5.c.g.99.14		14	7.3	odd	6
196.5.c.h.99.13		14	28.11	odd	6
196.5.c.h.99.14		14	7.4	even	3