Embedded newform 21.7.h.a.2.8

• Label: 21.7.h.a.2.8
• Level: $21$
• Weight: $7$
• Character: 21.2
• Analytic conductor: $4.831$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.83113575602\)
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			2.8
Character	\(\chi\)	\(=\)	21.2
Dual form			21.7.h.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.07584 + 0.621137i) q^{2} +(-9.77741 + 25.1675i) q^{3} +(-31.2284 - 54.0891i) q^{4} +(-93.7473 - 54.1250i) q^{5} +(-26.1514 + 21.0031i) q^{6} +(147.950 - 309.451i) q^{7} -157.094i q^{8} +(-537.804 - 492.146i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - q^{3} + 382 q^{4} - 356 q^{6} + 1120 q^{7} + 1031 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.07584	+	0.621137i	0.134480	+	0.0776422i	0.565731	−	0.824590i	\(-0.308594\pi\)
							−0.431251	+	0.902232i	\(0.641928\pi\)
\(3\)	−9.77741	+	25.1675i	−0.362126	+	0.932129i
							\(5\)	−93.7473	−	54.1250i	−0.749978	−	0.433000i	0.0757077	−	0.997130i	\(-0.475878\pi\)
−0.825686	+	0.564130i	\(0.809212\pi\)
\(7\)	147.950	−	309.451i	0.431341	−	0.902189i
							\(11\)	−1895.66	+	1094.46i	−1.42424	+	0.822283i	−0.996657	−	0.0816949i	\(-0.973967\pi\)
−0.427579	+	0.903978i	\(0.640633\pi\)
\(13\)	−1138.08			−0.518013			−0.259007	−	0.965876i	\(-0.583395\pi\)
							−0.259007	+	0.965876i	\(0.583395\pi\)
\(17\)	6949.27	−	4012.16i	1.41447	−	0.816643i	0.418661	−	0.908142i	\(-0.362499\pi\)
							0.995805	+	0.0914999i	\(0.0291661\pi\)
\(19\)	−3237.28	+	5607.13i	−0.471975	+	0.817485i	−0.999486	−	0.0320633i	\(-0.989792\pi\)
							0.527511	+	0.849548i	\(0.323126\pi\)
\(23\)	−1113.26	−	642.738i	−0.0914979	−	0.0528263i	0.453553	−	0.891229i	\(-0.350156\pi\)
							−0.545051	+	0.838403i	\(0.683490\pi\)
\(29\)		−	14229.8i		−	0.583450i	−0.956502	−	0.291725i	\(-0.905771\pi\)
							0.956502	−	0.291725i	\(-0.0942291\pi\)
\(31\)	−8129.75	−	14081.1i	−0.272893	−	0.472664i	0.696708	−	0.717354i	\(-0.254647\pi\)
							−0.969601	+	0.244690i	\(0.921314\pi\)
\(37\)	−16657.2	+	28851.0i	−0.328848	+	0.569582i	−0.982284	−	0.187400i	\(-0.939994\pi\)
							0.653435	+	0.756982i	\(0.273327\pi\)
\(41\)		−	98678.4i		−	1.43176i	−0.698223	−	0.715881i	\(-0.746025\pi\)
							0.698223	−	0.715881i	\(-0.253975\pi\)
\(43\)	62149.0			0.781680			0.390840	−	0.920459i	\(-0.372185\pi\)
							0.390840	+	0.920459i	\(0.372185\pi\)
\(47\)	50356.5	+	29073.3i	0.485023	+	0.280028i	0.722507	−	0.691363i	\(-0.242990\pi\)
							−0.237485	+	0.971391i	\(0.576323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.7.h.a.2.8	yes	28
3.2	odd	2	inner	21.7.h.a.2.7	✓	28
7.2	even	3		147.7.b.e.50.7		14
7.4	even	3	inner	21.7.h.a.11.7	yes	28
7.5	odd	6		147.7.b.d.50.7		14
21.2	odd	6		147.7.b.e.50.8		14
21.5	even	6		147.7.b.d.50.8		14
21.11	odd	6	inner	21.7.h.a.11.8	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.h.a.2.7	✓	28	3.2	odd	2	inner
21.7.h.a.2.8	yes	28	1.1	even	1	trivial
21.7.h.a.11.7	yes	28	7.4	even	3	inner
21.7.h.a.11.8	yes	28	21.11	odd	6	inner
147.7.b.d.50.7		14	7.5	odd	6
147.7.b.d.50.8		14	21.5	even	6
147.7.b.e.50.7		14	7.2	even	3
147.7.b.e.50.8		14	21.2	odd	6