Embedded newform 363.3.h.n.323.4

• Label: 363.3.h.n.323.4
• 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			323.4
Root			\(2.91048 - 0.945671i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.323
Dual form			363.3.h.n.245.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.79877 - 2.47580i) q^{2} +(-2.81156 - 1.04648i) q^{3} +(-1.65793 - 5.10257i) q^{4} +(3.90250 + 5.37133i) q^{5} +(-7.64823 + 5.07849i) q^{6} +(0.946512 + 2.91306i) q^{7} +(-3.97326 - 1.29099i) q^{8} +(6.80977 + 5.88447i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 5 q^{3} + 18 q^{4} - 32 q^{6} + 34 q^{7} + 17 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.79877	−	2.47580i	0.899386	−	1.23790i	−0.0712771	−	0.997457i	\(-0.522707\pi\)
							0.970663	−	0.240443i	\(-0.0772926\pi\)
\(3\)	−2.81156	−	1.04648i	−0.937188	−	0.348825i
							\(5\)	3.90250	+	5.37133i	0.780499	+	1.07427i	0.995227	+	0.0975910i	\(0.0311137\pi\)
−0.214727	+	0.976674i	\(0.568886\pi\)
\(7\)	0.946512	+	2.91306i	0.135216	+	0.416152i	0.995624	−	0.0934546i	\(-0.0297910\pi\)
							−0.860408	+	0.509607i	\(0.829791\pi\)
\(11\)		0			0
							\(13\)	13.1422	+	9.54837i	1.01094	+	0.734490i	0.964406	−	0.264427i	\(-0.0851827\pi\)
0.0465330	+	0.998917i	\(0.485183\pi\)
\(17\)	0.473548	+	0.651783i	0.0278557	+	0.0383401i	0.822718	−	0.568450i	\(-0.192457\pi\)
							−0.794862	+	0.606790i	\(0.792457\pi\)
\(19\)	−6.39720	+	19.6886i	−0.336695	+	1.03624i	0.629186	+	0.777255i	\(0.283388\pi\)
							−0.965881	+	0.258986i	\(0.916612\pi\)
\(23\)		−	27.3224i		−	1.18793i	−0.804491	−	0.593965i	\(-0.797562\pi\)
							0.804491	−	0.593965i	\(-0.202438\pi\)
\(29\)	3.59755	−	1.16891i	0.124053	−	0.0403074i	−0.246332	−	0.969185i	\(-0.579226\pi\)
							0.370386	+	0.928878i	\(0.379226\pi\)
\(31\)	16.8114	+	12.2142i	0.542302	+	0.394006i	0.824939	−	0.565221i	\(-0.191209\pi\)
							−0.282637	+	0.959227i	\(0.591209\pi\)
\(37\)	11.9137	+	36.6667i	0.321993	+	0.990991i	0.972780	+	0.231731i	\(0.0744390\pi\)
							−0.650787	+	0.759260i	\(0.725561\pi\)
\(41\)	−12.7181	−	4.13237i	−0.310199	−	0.100790i	0.149781	−	0.988719i	\(-0.452143\pi\)
							−0.459979	+	0.887930i	\(0.652143\pi\)
\(43\)	−43.4125			−1.00959			−0.504797	−	0.863238i	\(-0.668433\pi\)
							−0.504797	+	0.863238i	\(0.668433\pi\)
\(47\)	−18.9168	−	6.14644i	−0.402485	−	0.130775i	0.100777	−	0.994909i	\(-0.467867\pi\)
							−0.503262	+	0.864134i	\(0.667867\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.n.323.4		16
3.2	odd	2	inner	363.3.h.n.323.1		16
11.2	odd	10		33.3.h.b.5.4	yes	16
11.3	even	5	inner	363.3.h.n.245.1		16
11.4	even	5		363.3.h.j.251.4		16
11.5	even	5		363.3.b.l.122.7		8
11.6	odd	10		363.3.b.m.122.2		8
11.7	odd	10		33.3.h.b.20.1	yes	16
11.8	odd	10		363.3.h.o.245.4		16
11.9	even	5		363.3.h.j.269.1		16
11.10	odd	2		363.3.h.o.323.1		16
33.2	even	10		33.3.h.b.5.1	✓	16
33.5	odd	10		363.3.b.l.122.2		8
33.8	even	10		363.3.h.o.245.1		16
33.14	odd	10	inner	363.3.h.n.245.4		16
33.17	even	10		363.3.b.m.122.7		8
33.20	odd	10		363.3.h.j.269.4		16
33.26	odd	10		363.3.h.j.251.1		16
33.29	even	10		33.3.h.b.20.4	yes	16
33.32	even	2		363.3.h.o.323.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.5.1	✓	16	33.2	even	10
33.3.h.b.5.4	yes	16	11.2	odd	10
33.3.h.b.20.1	yes	16	11.7	odd	10
33.3.h.b.20.4	yes	16	33.29	even	10
363.3.b.l.122.2		8	33.5	odd	10
363.3.b.l.122.7		8	11.5	even	5
363.3.b.m.122.2		8	11.6	odd	10
363.3.b.m.122.7		8	33.17	even	10
363.3.h.j.251.1		16	33.26	odd	10
363.3.h.j.251.4		16	11.4	even	5
363.3.h.j.269.1		16	11.9	even	5
363.3.h.j.269.4		16	33.20	odd	10
363.3.h.n.245.1		16	11.3	even	5	inner
363.3.h.n.245.4		16	33.14	odd	10	inner
363.3.h.n.323.1		16	3.2	odd	2	inner
363.3.h.n.323.4		16	1.1	even	1	trivial
363.3.h.o.245.1		16	33.8	even	10
363.3.h.o.245.4		16	11.8	odd	10
363.3.h.o.323.1		16	11.10	odd	2
363.3.h.o.323.4		16	33.32	even	2