Embedded newform 363.3.h.m.251.1

• Label: 363.3.h.m.251.1
• Level: $363$
• Weight: $3$
• Character: 363.251
• Analytic conductor: $9.891$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $8$

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:	16.0.343361479062744140625.1
Defining polynomial:	x^16 - x^15 + 3*x^14 - 8*x^13 + 8*x^12 + 7*x^11 + 6*x^10 + 56*x^9 - 137*x^8 + 168*x^7 + 54*x^6 + 189*x^5 + 648*x^4 - 1944*x^3 + 2187*x^2 - 2187*x + 6561
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			251.1
Root			\(1.56693 + 0.738055i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.251
Dual form			363.3.h.m.269.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.40079 - 0.780063i) q^{2} +(2.95836 - 0.498102i) q^{3} +(1.91922 + 1.39439i) q^{4} +(0.753510 - 0.244830i) q^{5} +(-7.49095 - 1.11187i) q^{6} +(-5.45647 - 3.96435i) q^{7} +(2.41516 + 3.32418i) q^{8} +(8.50379 - 2.94713i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 5 q^{3} - 2 q^{4} + 2 q^{6} - 4 q^{7} + 7 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(244\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.40079	−	0.780063i	−1.20039	−	0.390032i	−0.360488	−	0.932764i	\(-0.617390\pi\)
							−0.839906	+	0.542732i	\(0.817390\pi\)
\(3\)	2.95836	−	0.498102i	0.986120	−	0.166034i
							\(5\)	0.753510	−	0.244830i	0.150702	−	0.0489660i	−0.232695	−	0.972550i	\(-0.574754\pi\)
0.383397	+	0.923584i	\(0.374754\pi\)
\(7\)	−5.45647	−	3.96435i	−0.779495	−	0.566336i	0.125332	−	0.992115i	\(-0.460000\pi\)
							−0.904827	+	0.425778i	\(0.860000\pi\)
\(11\)		0			0
							\(13\)	2.93230	−	9.02469i	0.225562	−	0.694207i	−0.772673	−	0.634805i	\(-0.781080\pi\)
0.998234	−	0.0594024i	\(-0.0189195\pi\)
\(17\)	−27.8635	+	9.05339i	−1.63903	+	0.532552i	−0.976321	−	0.216327i	\(-0.930592\pi\)
							−0.662707	+	0.748879i	\(0.730592\pi\)
\(19\)	21.2235	−	15.4198i	1.11703	−	0.811567i	0.133271	−	0.991080i	\(-0.457452\pi\)
							0.983756	+	0.179513i	\(0.0574521\pi\)
\(23\)		−	26.9205i		−	1.17046i	−0.810868	−	0.585229i	\(-0.801005\pi\)
							0.810868	−	0.585229i	\(-0.198995\pi\)
\(29\)	15.2490	−	20.9884i	0.525826	−	0.723738i	−0.460661	−	0.887576i	\(-0.652388\pi\)
							0.986487	+	0.163838i	\(0.0523875\pi\)
\(31\)	−0.884223	+	2.72136i	−0.0285233	+	0.0877858i	−0.964305	−	0.264795i	\(-0.914696\pi\)
							0.935781	+	0.352581i	\(0.114696\pi\)
\(37\)	1.93681	+	1.40718i	0.0523463	+	0.0380318i	0.613651	−	0.789578i	\(-0.289700\pi\)
							−0.561304	+	0.827609i	\(0.689700\pi\)
\(41\)	−10.3541	−	14.2512i	−0.252539	−	0.347590i	0.663859	−	0.747857i	\(-0.268917\pi\)
							−0.916399	+	0.400267i	\(0.868917\pi\)
\(43\)	12.5109			0.290951			0.145475	−	0.989362i	\(-0.453529\pi\)
							0.145475	+	0.989362i	\(0.453529\pi\)
\(47\)	−24.4983	−	33.7190i	−0.521241	−	0.717426i	0.464523	−	0.885561i	\(-0.346226\pi\)
							−0.985764	+	0.168135i	\(0.946226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.m.251.1		16
3.2	odd	2	inner	363.3.h.m.251.4		16
11.2	odd	10		363.3.h.l.245.1		16
11.3	even	5	inner	363.3.h.m.323.1		16
11.4	even	5		33.3.b.b.23.1	✓	4
11.5	even	5	inner	363.3.h.m.269.4		16
11.6	odd	10		363.3.h.l.269.1		16
11.7	odd	10		363.3.b.h.122.4		4
11.8	odd	10		363.3.h.l.323.4		16
11.9	even	5	inner	363.3.h.m.245.4		16
11.10	odd	2		363.3.h.l.251.4		16
33.2	even	10		363.3.h.l.245.4		16
33.5	odd	10	inner	363.3.h.m.269.1		16
33.8	even	10		363.3.h.l.323.1		16
33.14	odd	10	inner	363.3.h.m.323.4		16
33.17	even	10		363.3.h.l.269.4		16
33.20	odd	10	inner	363.3.h.m.245.1		16
33.26	odd	10		33.3.b.b.23.4	yes	4
33.29	even	10		363.3.b.h.122.1		4
33.32	even	2		363.3.h.l.251.1		16
44.15	odd	10		528.3.i.d.353.4		4
132.59	even	10		528.3.i.d.353.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.b.b.23.1	✓	4	11.4	even	5
33.3.b.b.23.4	yes	4	33.26	odd	10
363.3.b.h.122.1		4	33.29	even	10
363.3.b.h.122.4		4	11.7	odd	10
363.3.h.l.245.1		16	11.2	odd	10
363.3.h.l.245.4		16	33.2	even	10
363.3.h.l.251.1		16	33.32	even	2
363.3.h.l.251.4		16	11.10	odd	2
363.3.h.l.269.1		16	11.6	odd	10
363.3.h.l.269.4		16	33.17	even	10
363.3.h.l.323.1		16	33.8	even	10
363.3.h.l.323.4		16	11.8	odd	10
363.3.h.m.245.1		16	33.20	odd	10	inner
363.3.h.m.245.4		16	11.9	even	5	inner
363.3.h.m.251.1		16	1.1	even	1	trivial
363.3.h.m.251.4		16	3.2	odd	2	inner
363.3.h.m.269.1		16	33.5	odd	10	inner
363.3.h.m.269.4		16	11.5	even	5	inner
363.3.h.m.323.1		16	11.3	even	5	inner
363.3.h.m.323.4		16	33.14	odd	10	inner
528.3.i.d.353.3		4	132.59	even	10
528.3.i.d.353.4		4	44.15	odd	10