Embedded newform 28.5.g.a.11.1

• Label: 28.5.g.a.11.1
• Level: $28$
• Weight: $5$
• Character: 28.11
• Analytic conductor: $2.894$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• 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			11.1
Character	\(\chi\)	\(=\)	28.11
Dual form			28.5.g.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98249 + 0.373846i) q^{2} +(-10.4527 - 6.03487i) q^{3} +(15.7205 - 2.97768i) q^{4} +(13.8853 + 24.0501i) q^{5} +(43.8839 + 20.1261i) q^{6} +(34.1259 + 35.1628i) q^{7} +(-61.4935 + 17.7356i) q^{8} +(32.3394 + 56.0135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{2} - 4 q^{4} - 2 q^{5} - 36 q^{6} - 184 q^{8} + 268 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98249	+	0.373846i	−0.995623	+	0.0934616i
							\(3\)	−10.4527	−	6.03487i	−1.16141	−	0.670542i	−0.209770	−	0.977751i	\(-0.567272\pi\)
−0.951642	+	0.307209i	\(0.900605\pi\)
\(5\)	13.8853	+	24.0501i	0.555412	+	0.962002i	0.997871	+	0.0652135i	\(0.0207729\pi\)
							−0.442459	+	0.896789i	\(0.645894\pi\)
\(7\)	34.1259	+	35.1628i	0.696446	+	0.717609i
							\(11\)	91.7023	+	52.9444i	0.757871	+	0.437557i	0.828531	−	0.559944i	\(-0.189177\pi\)
−0.0706601	+	0.997500i	\(0.522511\pi\)
\(13\)	111.726			0.661103			0.330552	−	0.943788i	\(-0.392765\pi\)
							0.330552	+	0.943788i	\(0.392765\pi\)
\(17\)	−268.141	+	464.434i	−0.927823	+	1.60704i	−0.140867	+	0.990029i	\(0.544989\pi\)
							−0.786957	+	0.617008i	\(0.788344\pi\)
\(19\)	−15.8167	+	9.13175i	−0.0438135	+	0.0252957i	−0.521747	−	0.853100i	\(-0.674719\pi\)
							0.477933	+	0.878396i	\(0.341386\pi\)
\(23\)	−246.453	+	142.290i	−0.465885	+	0.268979i	−0.714515	−	0.699620i	\(-0.753353\pi\)
							0.248631	+	0.968598i	\(0.420019\pi\)
\(29\)	1348.88			1.60389			0.801947	−	0.597395i	\(-0.203797\pi\)
							0.801947	+	0.597395i	\(0.203797\pi\)
\(31\)	418.793	+	241.790i	0.435789	+	0.251603i	0.701810	−	0.712364i	\(-0.252376\pi\)
							−0.266021	+	0.963967i	\(0.585709\pi\)
\(37\)	−807.077	−	1397.90i	−0.589537	−	1.02111i	−0.994293	−	0.106684i	\(-0.965977\pi\)
							0.404756	−	0.914425i	\(-0.367357\pi\)
\(41\)	−209.077			−0.124376			−0.0621882	−	0.998064i	\(-0.519808\pi\)
							−0.0621882	+	0.998064i	\(0.519808\pi\)
\(43\)		−	2072.51i		−	1.12088i	−0.828194	−	0.560442i	\(-0.810631\pi\)
							0.828194	−	0.560442i	\(-0.189369\pi\)
\(47\)	−1737.55	+	1003.17i	−0.786577	+	0.454130i	−0.838756	−	0.544507i	\(-0.816717\pi\)
							0.0521793	+	0.998638i	\(0.483383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.5.g.a.11.1	✓	28
4.3	odd	2	inner	28.5.g.a.11.10	yes	28
7.2	even	3	inner	28.5.g.a.23.10	yes	28
7.3	odd	6		196.5.c.g.99.9		14
7.4	even	3		196.5.c.h.99.9		14
28.3	even	6		196.5.c.g.99.10		14
28.11	odd	6		196.5.c.h.99.10		14
28.23	odd	6	inner	28.5.g.a.23.1	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.g.a.11.1	✓	28	1.1	even	1	trivial
28.5.g.a.11.10	yes	28	4.3	odd	2	inner
28.5.g.a.23.1	yes	28	28.23	odd	6	inner
28.5.g.a.23.10	yes	28	7.2	even	3	inner
196.5.c.g.99.9		14	7.3	odd	6
196.5.c.g.99.10		14	28.3	even	6
196.5.c.h.99.9		14	7.4	even	3
196.5.c.h.99.10		14	28.11	odd	6