Embedded newform 363.3.h.j.245.4

• Label: 363.3.h.j.245.4
• Level: $363$
• Weight: $3$
• Character: 363.245
• 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			245.4
Root			\(2.10855 + 2.90217i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.245
Dual form			363.3.h.j.323.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10855 + 2.90217i) q^{2} +(0.307087 - 2.98424i) q^{3} +(-2.74053 + 8.43448i) q^{4} +(-1.22635 + 1.68793i) q^{5} +(9.30827 - 5.40120i) q^{6} +(-2.73883 + 8.42924i) q^{7} +(-16.6100 + 5.39692i) q^{8} +(-8.81140 - 1.83284i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{3} + 8 q^{4} + 33 q^{6} - 6 q^{7} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10855	+	2.90217i	1.05427	+	1.45108i	0.885045	+	0.465506i	\(0.154128\pi\)
							0.169229	+	0.985577i	\(0.445872\pi\)
\(3\)	0.307087	−	2.98424i	0.102362	−	0.994747i
							\(5\)	−1.22635	+	1.68793i	−0.245270	+	0.337586i	−0.913848	−	0.406057i	\(-0.866903\pi\)
0.668578	+	0.743642i	\(0.266903\pi\)
\(7\)	−2.73883	+	8.42924i	−0.391261	+	1.20418i	0.540575	+	0.841296i	\(0.318207\pi\)
							−0.931836	+	0.362881i	\(0.881793\pi\)
\(11\)		0			0
							\(13\)	−1.33068	+	0.966792i	−0.102360	+	0.0743686i	−0.637788	−	0.770212i	\(-0.720150\pi\)
0.535428	+	0.844581i	\(0.320150\pi\)
\(17\)	−7.30235	+	10.0508i	−0.429550	+	0.591225i	−0.967850	−	0.251529i	\(-0.919067\pi\)
							0.538300	+	0.842753i	\(0.319067\pi\)
\(19\)	−3.26497	−	10.0485i	−0.171841	−	0.528871i	0.827635	−	0.561267i	\(-0.189686\pi\)
							−0.999475	+	0.0323966i	\(0.989686\pi\)
\(23\)			20.3378i			0.884251i	0.896953	+	0.442126i	\(0.145775\pi\)
							−0.896953	+	0.442126i	\(0.854225\pi\)
\(29\)	−11.0405	−	3.58727i	−0.380707	−	0.123699i	0.112411	−	0.993662i	\(-0.464143\pi\)
							−0.493118	+	0.869963i	\(0.664143\pi\)
\(31\)	18.9215	−	13.7473i	0.610371	−	0.443460i	−0.239174	−	0.970977i	\(-0.576877\pi\)
							0.849545	+	0.527516i	\(0.176877\pi\)
\(37\)	2.23911	−	6.89128i	0.0605166	−	0.186251i	−0.916228	−	0.400657i	\(-0.868782\pi\)
							0.976745	+	0.214407i	\(0.0687817\pi\)
\(41\)	36.9741	−	12.0136i	0.901806	−	0.293015i	0.178824	−	0.983881i	\(-0.442771\pi\)
							0.722982	+	0.690866i	\(0.242771\pi\)
\(43\)	15.8444			0.368474			0.184237	−	0.982882i	\(-0.441019\pi\)
							0.184237	+	0.982882i	\(0.441019\pi\)
\(47\)	−43.0910	+	14.0011i	−0.916831	+	0.297896i	−0.729166	−	0.684337i	\(-0.760092\pi\)
							−0.187665	+	0.982233i	\(0.560092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.j.245.4		16
3.2	odd	2	inner	363.3.h.j.245.1		16
11.2	odd	10		363.3.b.m.122.8		8
11.3	even	5		363.3.h.n.269.4		16
11.4	even	5	inner	363.3.h.j.323.1		16
11.5	even	5		363.3.h.n.251.1		16
11.6	odd	10		363.3.h.o.251.4		16
11.7	odd	10		33.3.h.b.26.4	yes	16
11.8	odd	10		363.3.h.o.269.1		16
11.9	even	5		363.3.b.l.122.1		8
11.10	odd	2		33.3.h.b.14.1	✓	16
33.2	even	10		363.3.b.m.122.1		8
33.5	odd	10		363.3.h.n.251.4		16
33.8	even	10		363.3.h.o.269.4		16
33.14	odd	10		363.3.h.n.269.1		16
33.17	even	10		363.3.h.o.251.1		16
33.20	odd	10		363.3.b.l.122.8		8
33.26	odd	10	inner	363.3.h.j.323.4		16
33.29	even	10		33.3.h.b.26.1	yes	16
33.32	even	2		33.3.h.b.14.4	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.14.1	✓	16	11.10	odd	2
33.3.h.b.14.4	yes	16	33.32	even	2
33.3.h.b.26.1	yes	16	33.29	even	10
33.3.h.b.26.4	yes	16	11.7	odd	10
363.3.b.l.122.1		8	11.9	even	5
363.3.b.l.122.8		8	33.20	odd	10
363.3.b.m.122.1		8	33.2	even	10
363.3.b.m.122.8		8	11.2	odd	10
363.3.h.j.245.1		16	3.2	odd	2	inner
363.3.h.j.245.4		16	1.1	even	1	trivial
363.3.h.j.323.1		16	11.4	even	5	inner
363.3.h.j.323.4		16	33.26	odd	10	inner
363.3.h.n.251.1		16	11.5	even	5
363.3.h.n.251.4		16	33.5	odd	10
363.3.h.n.269.1		16	33.14	odd	10
363.3.h.n.269.4		16	11.3	even	5
363.3.h.o.251.1		16	33.17	even	10
363.3.h.o.251.4		16	11.6	odd	10
363.3.h.o.269.1		16	11.8	odd	10
363.3.h.o.269.4		16	33.8	even	10