Embedded newform 21.5.h.b.2.5

• Label: 21.5.h.b.2.5
• Level: $21$
• Weight: $5$
• Character: 21.2
• Analytic conductor: $2.171$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17076922476\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 103*x^14 + 7227*x^12 - 270898*x^10 + 7374256*x^8 - 115494792*x^6 + 1245573504*x^4 - 2908017504*x^2 + 5639409216
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2.5
Root			\(1.34450 + 0.776250i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.2
Dual form			21.5.h.b.11.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34450 + 0.776250i) q^{2} +(8.38348 - 3.27372i) q^{3} +(-6.79487 - 11.7691i) q^{4} +(23.4142 + 13.5182i) q^{5} +(13.8128 + 2.10615i) q^{6} +(-27.0847 + 40.8340i) q^{7} -45.9381i q^{8} +(59.5656 - 54.8903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} + 78 q^{4} + 28 q^{6} - 168 q^{7} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34450	+	0.776250i	0.336126	+	0.194062i	0.658558	−	0.752530i	\(-0.271167\pi\)
							−0.322432	+	0.946593i	\(0.604500\pi\)
\(3\)	8.38348	−	3.27372i	0.931498	−	0.363746i
							\(5\)	23.4142	+	13.5182i	0.936569	+	0.540728i	0.888883	−	0.458134i	\(-0.151482\pi\)
0.0476860	+	0.998862i	\(0.484815\pi\)
\(7\)	−27.0847	+	40.8340i	−0.552749	+	0.833348i
							\(11\)	−130.154	+	75.1445i	−1.07565	+	0.621029i	−0.929720	−	0.368266i	\(-0.879952\pi\)
−0.145933	+	0.989295i	\(0.546618\pi\)
\(13\)	−189.322			−1.12025			−0.560126	−	0.828408i	\(-0.689247\pi\)
							−0.560126	+	0.828408i	\(0.689247\pi\)
\(17\)	136.932	−	79.0575i	0.473812	−	0.273555i	−0.244022	−	0.969770i	\(-0.578467\pi\)
							0.717834	+	0.696214i	\(0.245134\pi\)
\(19\)	67.5003	−	116.914i	0.186981	−	0.323861i	−0.757261	−	0.653112i	\(-0.773463\pi\)
							0.944242	+	0.329251i	\(0.106796\pi\)
\(23\)	791.290	+	456.852i	1.49582	+	0.863614i	0.999988	−	0.00480324i	\(-0.00152892\pi\)
							0.495835	+	0.868417i	\(0.334862\pi\)
\(29\)		−	177.015i		−	0.210481i	−0.994447	−	0.105241i	\(-0.966439\pi\)
							0.994447	−	0.105241i	\(-0.0335613\pi\)
\(31\)	−134.801	−	233.483i	−0.140272	−	0.242958i	0.787327	−	0.616536i	\(-0.211464\pi\)
							−0.927599	+	0.373577i	\(0.878131\pi\)
\(37\)	205.859	−	356.558i	0.150372	−	0.260451i	−0.780992	−	0.624540i	\(-0.785286\pi\)
							0.931364	+	0.364089i	\(0.118620\pi\)
\(41\)		−	1725.66i		−	1.02657i	−0.858219	−	0.513283i	\(-0.828429\pi\)
							0.858219	−	0.513283i	\(-0.171571\pi\)
\(43\)	811.064			0.438650			0.219325	−	0.975652i	\(-0.429615\pi\)
							0.219325	+	0.975652i	\(0.429615\pi\)
\(47\)	−211.453	−	122.082i	−0.0957234	−	0.0552659i	0.451374	−	0.892335i	\(-0.350934\pi\)
							−0.547098	+	0.837069i	\(0.684267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.h.b.2.5	yes	16
3.2	odd	2	inner	21.5.h.b.2.4	✓	16
7.2	even	3		147.5.b.c.50.4		8
7.3	odd	6		147.5.h.b.116.4		16
7.4	even	3	inner	21.5.h.b.11.4	yes	16
7.5	odd	6		147.5.b.f.50.4		8
7.6	odd	2		147.5.h.b.128.5		16
21.2	odd	6		147.5.b.c.50.5		8
21.5	even	6		147.5.b.f.50.5		8
21.11	odd	6	inner	21.5.h.b.11.5	yes	16
21.17	even	6		147.5.h.b.116.5		16
21.20	even	2		147.5.h.b.128.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.h.b.2.4	✓	16	3.2	odd	2	inner
21.5.h.b.2.5	yes	16	1.1	even	1	trivial
21.5.h.b.11.4	yes	16	7.4	even	3	inner
21.5.h.b.11.5	yes	16	21.11	odd	6	inner
147.5.b.c.50.4		8	7.2	even	3
147.5.b.c.50.5		8	21.2	odd	6
147.5.b.f.50.4		8	7.5	odd	6
147.5.b.f.50.5		8	21.5	even	6
147.5.h.b.116.4		16	7.3	odd	6
147.5.h.b.116.5		16	21.17	even	6
147.5.h.b.128.4		16	21.20	even	2
147.5.h.b.128.5		16	7.6	odd	2