Embedded newform 363.3.h.j.323.2

• Label: 363.3.h.j.323.2
• Level: $363$
• Weight: $3$
• Character: 363.323
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	363.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89103359628\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			323.2
Root			\(-0.974642 + 1.34148i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.323
Dual form			363.3.h.j.245.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.974642 + 1.34148i) q^{2} +(-1.09751 - 2.79204i) q^{3} +(0.386428 + 1.18930i) q^{4} +(-0.410570 - 0.565101i) q^{5} +(4.81514 + 1.24895i) q^{6} +(-0.806259 - 2.48141i) q^{7} +(-8.28007 - 2.69036i) q^{8} +(-6.59095 + 6.12857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{3} + 8 q^{4} + 33 q^{6} - 6 q^{7} - 28 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.974642	+	1.34148i	−0.487321	+	0.670740i	−0.979891	−	0.199533i	\(-0.936057\pi\)
							0.492570	+	0.870273i	\(0.336057\pi\)
\(3\)	−1.09751	−	2.79204i	−0.365836	−	0.930679i
							\(5\)	−0.410570	−	0.565101i	−0.0821140	−	0.113020i	0.765984	−	0.642860i	\(-0.222252\pi\)
−0.848098	+	0.529839i	\(0.822252\pi\)
\(7\)	−0.806259	−	2.48141i	−0.115180	−	0.354487i	0.876805	−	0.480847i	\(-0.159671\pi\)
							−0.991985	+	0.126360i	\(0.959671\pi\)
\(11\)		0			0
							\(13\)	13.8479	+	10.0611i	1.06522	+	0.773930i	0.975048	−	0.221996i	\(-0.0712573\pi\)
0.0901753	+	0.995926i	\(0.471257\pi\)
\(17\)	9.47072	+	13.0353i	0.557101	+	0.766784i	0.990954	−	0.134200i	\(-0.0428464\pi\)
							−0.433853	+	0.900984i	\(0.642846\pi\)
\(19\)	4.92151	−	15.1469i	0.259027	−	0.797203i	−0.733983	−	0.679168i	\(-0.762341\pi\)
							0.993009	−	0.118035i	\(-0.0376594\pi\)
\(23\)		−	23.1295i		−	1.00563i	−0.864395	−	0.502814i	\(-0.832298\pi\)
							0.864395	−	0.502814i	\(-0.167702\pi\)
\(29\)	5.10329	−	1.65816i	0.175976	−	0.0571779i	−0.219704	−	0.975567i	\(-0.570509\pi\)
							0.395679	+	0.918389i	\(0.370509\pi\)
\(31\)	3.28671	+	2.38793i	0.106023	+	0.0770301i	0.639533	−	0.768763i	\(-0.279128\pi\)
							−0.533511	+	0.845793i	\(0.679128\pi\)
\(37\)	19.6322	+	60.4217i	0.530600	+	1.63302i	0.752968	+	0.658057i	\(0.228621\pi\)
							−0.222368	+	0.974963i	\(0.571379\pi\)
\(41\)	64.1371	+	20.8394i	1.56432	+	0.508278i	0.957958	−	0.286910i	\(-0.0926281\pi\)
							0.606362	+	0.795188i	\(0.292628\pi\)
\(43\)	22.6622			0.527028			0.263514	−	0.964656i	\(-0.415118\pi\)
							0.263514	+	0.964656i	\(0.415118\pi\)
\(47\)	70.2078	+	22.8119i	1.49378	+	0.485360i	0.938198	−	0.346100i	\(-0.112494\pi\)
							0.555585	+	0.831460i	\(0.312494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.j.323.2		16
3.2	odd	2	inner	363.3.h.j.323.3		16
11.2	odd	10		363.3.h.o.269.2		16
11.3	even	5	inner	363.3.h.j.245.3		16
11.4	even	5		363.3.h.n.251.2		16
11.5	even	5		363.3.b.l.122.4		8
11.6	odd	10		363.3.b.m.122.5		8
11.7	odd	10		363.3.h.o.251.3		16
11.8	odd	10		33.3.h.b.14.2	✓	16
11.9	even	5		363.3.h.n.269.3		16
11.10	odd	2		33.3.h.b.26.3	yes	16
33.2	even	10		363.3.h.o.269.3		16
33.5	odd	10		363.3.b.l.122.5		8
33.8	even	10		33.3.h.b.14.3	yes	16
33.14	odd	10	inner	363.3.h.j.245.2		16
33.17	even	10		363.3.b.m.122.4		8
33.20	odd	10		363.3.h.n.269.2		16
33.26	odd	10		363.3.h.n.251.3		16
33.29	even	10		363.3.h.o.251.2		16
33.32	even	2		33.3.h.b.26.2	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.14.2	✓	16	11.8	odd	10
33.3.h.b.14.3	yes	16	33.8	even	10
33.3.h.b.26.2	yes	16	33.32	even	2
33.3.h.b.26.3	yes	16	11.10	odd	2
363.3.b.l.122.4		8	11.5	even	5
363.3.b.l.122.5		8	33.5	odd	10
363.3.b.m.122.4		8	33.17	even	10
363.3.b.m.122.5		8	11.6	odd	10
363.3.h.j.245.2		16	33.14	odd	10	inner
363.3.h.j.245.3		16	11.3	even	5	inner
363.3.h.j.323.2		16	1.1	even	1	trivial
363.3.h.j.323.3		16	3.2	odd	2	inner
363.3.h.n.251.2		16	11.4	even	5
363.3.h.n.251.3		16	33.26	odd	10
363.3.h.n.269.2		16	33.20	odd	10
363.3.h.n.269.3		16	11.9	even	5
363.3.h.o.251.2		16	33.29	even	10
363.3.h.o.251.3		16	11.7	odd	10
363.3.h.o.269.2		16	11.2	odd	10
363.3.h.o.269.3		16	33.2	even	10