Embedded newform 28.6.f.a.3.8

• Label: 28.6.f.a.3.8
• Level: $28$
• Weight: $6$
• Character: 28.3
• 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			3.8
Character	\(\chi\)	\(=\)	28.3
Dual form			28.6.f.a.19.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.987129 - 5.57006i) q^{2} +(-10.6204 + 18.3950i) q^{3} +(-30.0512 + 10.9967i) q^{4} +(88.2183 - 50.9329i) q^{5} +(112.945 + 40.9978i) q^{6} +(128.565 - 16.6726i) q^{7} +(90.9169 + 156.532i) q^{8} +(-104.084 - 180.280i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.987129	−	5.57006i	−0.174501	−	0.984657i
							\(3\)	−10.6204	+	18.3950i	−0.681297	+	1.18004i	0.293288	+	0.956024i	\(0.405250\pi\)
−0.974585	+	0.224017i	\(0.928083\pi\)
\(5\)	88.2183	−	50.9329i	1.57810	−	0.911115i	0.582972	−	0.812492i	\(-0.301890\pi\)
							0.995125	−	0.0986224i	\(-0.0314436\pi\)
\(7\)	128.565	−	16.6726i	0.991696	−	0.128605i
							\(11\)	265.056	+	153.030i	0.660474	+	0.381325i	0.792457	−	0.609927i	\(-0.208801\pi\)
−0.131984	+	0.991252i	\(0.542135\pi\)
\(13\)			77.2446i			0.126768i	0.997989	+	0.0633840i	\(0.0201893\pi\)
							−0.997989	+	0.0633840i	\(0.979811\pi\)
\(17\)	870.050	+	502.323i	0.730166	+	0.421562i	0.818483	−	0.574531i	\(-0.194815\pi\)
							−0.0883168	+	0.996092i	\(0.528149\pi\)
\(19\)	−582.760	−	1009.37i	−0.370344	−	0.641455i	0.619274	−	0.785175i	\(-0.287427\pi\)
							−0.989618	+	0.143720i	\(0.954094\pi\)
\(23\)	61.7004	−	35.6227i	0.0243203	−	0.0140413i	−0.487791	−	0.872961i	\(-0.662197\pi\)
							0.512111	+	0.858919i	\(0.328864\pi\)
\(29\)	−1666.73			−0.368019			−0.184009	−	0.982924i	\(-0.558908\pi\)
							−0.184009	+	0.982924i	\(0.558908\pi\)
\(31\)	−2516.99	+	4359.56i	−0.470412	+	0.814777i	−0.999427	−	0.0338351i	\(-0.989228\pi\)
							0.529016	+	0.848612i	\(0.322561\pi\)
\(37\)	4018.89	+	6960.92i	0.482616	+	0.835916i	0.999801	−	0.0199582i	\(-0.00635333\pi\)
							−0.517185	+	0.855874i	\(0.673020\pi\)
\(41\)		−	2605.51i		−	0.242066i	−0.992649	−	0.121033i	\(-0.961379\pi\)
							0.992649	−	0.121033i	\(-0.0386206\pi\)
\(43\)		−	5095.51i		−	0.420258i	−0.977674	−	0.210129i	\(-0.932612\pi\)
							0.977674	−	0.210129i	\(-0.0673884\pi\)
\(47\)	−13776.3	−	23861.3i	−0.909679	−	1.57561i	−0.814510	−	0.580149i	\(-0.802994\pi\)
							−0.0951689	−	0.995461i	\(-0.530339\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.6.f.a.3.8	yes	36
4.3	odd	2	inner	28.6.f.a.3.5	✓	36
7.3	odd	6		196.6.d.b.195.35		36
7.4	even	3		196.6.d.b.195.36		36
7.5	odd	6	inner	28.6.f.a.19.5	yes	36
28.3	even	6		196.6.d.b.195.34		36
28.11	odd	6		196.6.d.b.195.33		36
28.19	even	6	inner	28.6.f.a.19.8	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.5	✓	36	4.3	odd	2	inner
28.6.f.a.3.8	yes	36	1.1	even	1	trivial
28.6.f.a.19.5	yes	36	7.5	odd	6	inner
28.6.f.a.19.8	yes	36	28.19	even	6	inner
196.6.d.b.195.33		36	28.11	odd	6
196.6.d.b.195.34		36	28.3	even	6
196.6.d.b.195.35		36	7.3	odd	6
196.6.d.b.195.36		36	7.4	even	3