Embedded newform 363.3.h.l.245.1

• Label: 363.3.h.l.245.1
• Level: $363$
• Weight: $3$
• Character: 363.245
• 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			245.1
Root			\(-0.217724 - 1.71831i\) of defining polynomial
Character	\(\chi\)	\(=\)	363.245
Dual form			363.3.h.l.323.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48377 - 2.04223i) q^{2} +(0.440460 - 2.96749i) q^{3} +(-0.733075 + 2.25617i) q^{4} +(-0.465695 + 0.640974i) q^{5} +(-6.71384 + 3.50355i) q^{6} +(-2.08418 + 6.41446i) q^{7} +(-3.90781 + 1.26972i) q^{8} +(-8.61199 - 2.61412i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 5 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 7 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48377	−	2.04223i	−0.741884	−	1.02112i	−0.998508	−	0.0546047i	\(-0.982610\pi\)
							0.256624	−	0.966511i	\(-0.417390\pi\)
\(3\)	0.440460	−	2.96749i	0.146820	−	0.989163i
							\(5\)	−0.465695	+	0.640974i	−0.0931389	+	0.128195i	−0.853043	−	0.521840i	\(-0.825246\pi\)
0.759904	+	0.650035i	\(0.225246\pi\)
\(7\)	−2.08418	+	6.41446i	−0.297741	+	0.916351i	0.684546	+	0.728969i	\(0.260000\pi\)
							−0.982287	+	0.187382i	\(0.940000\pi\)
\(11\)		0			0
							\(13\)	7.67686	−	5.57757i	0.590528	−	0.429044i	−0.251976	−	0.967733i	\(-0.581080\pi\)
0.842504	+	0.538690i	\(0.181080\pi\)
\(17\)	−17.2206	+	23.7021i	−1.01297	+	1.39424i	−0.0959609	+	0.995385i	\(0.530592\pi\)
							−0.917014	+	0.398855i	\(0.869408\pi\)
\(19\)	8.10666	+	24.9497i	0.426666	+	1.31314i	0.901390	+	0.433008i	\(0.142548\pi\)
							−0.474724	+	0.880135i	\(0.657452\pi\)
\(23\)		−	26.9205i		−	1.17046i	−0.810868	−	0.585229i	\(-0.801005\pi\)
							0.810868	−	0.585229i	\(-0.198995\pi\)
\(29\)	−24.6733	−	8.01686i	−0.850805	−	0.276443i	−0.149022	−	0.988834i	\(-0.547612\pi\)
							−0.701783	+	0.712391i	\(0.747612\pi\)
\(31\)	2.31493	−	1.68189i	0.0746751	−	0.0542546i	−0.549821	−	0.835282i	\(-0.685304\pi\)
							0.624496	+	0.781028i	\(0.285304\pi\)
\(37\)	−0.739796	+	2.27686i	−0.0199945	+	0.0615367i	−0.960556	−	0.278087i	\(-0.910300\pi\)
							0.940561	+	0.339624i	\(0.110300\pi\)
\(41\)	16.7533	−	5.44348i	0.408617	−	0.132768i	−0.0974939	−	0.995236i	\(-0.531083\pi\)
							0.506111	+	0.862468i	\(0.331083\pi\)
\(43\)	−12.5109			−0.290951			−0.145475	−	0.989362i	\(-0.546471\pi\)
							−0.145475	+	0.989362i	\(0.546471\pi\)
\(47\)	−39.6391	+	12.8795i	−0.843385	+	0.274032i	−0.698673	−	0.715441i	\(-0.746226\pi\)
							−0.144712	+	0.989474i	\(0.546226\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.3.h.l.245.1		16
3.2	odd	2	inner	363.3.h.l.245.4		16
11.2	odd	10		33.3.b.b.23.1	✓	4
11.3	even	5	inner	363.3.h.l.269.1		16
11.4	even	5	inner	363.3.h.l.323.4		16
11.5	even	5	inner	363.3.h.l.251.4		16
11.6	odd	10		363.3.h.m.251.1		16
11.7	odd	10		363.3.h.m.323.1		16
11.8	odd	10		363.3.h.m.269.4		16
11.9	even	5		363.3.b.h.122.4		4
11.10	odd	2		363.3.h.m.245.4		16
33.2	even	10		33.3.b.b.23.4	yes	4
33.5	odd	10	inner	363.3.h.l.251.1		16
33.8	even	10		363.3.h.m.269.1		16
33.14	odd	10	inner	363.3.h.l.269.4		16
33.17	even	10		363.3.h.m.251.4		16
33.20	odd	10		363.3.b.h.122.1		4
33.26	odd	10	inner	363.3.h.l.323.1		16
33.29	even	10		363.3.h.m.323.4		16
33.32	even	2		363.3.h.m.245.1		16
44.35	even	10		528.3.i.d.353.4		4
132.35	odd	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	11.2	odd	10
33.3.b.b.23.4	yes	4	33.2	even	10
363.3.b.h.122.1		4	33.20	odd	10
363.3.b.h.122.4		4	11.9	even	5
363.3.h.l.245.1		16	1.1	even	1	trivial
363.3.h.l.245.4		16	3.2	odd	2	inner
363.3.h.l.251.1		16	33.5	odd	10	inner
363.3.h.l.251.4		16	11.5	even	5	inner
363.3.h.l.269.1		16	11.3	even	5	inner
363.3.h.l.269.4		16	33.14	odd	10	inner
363.3.h.l.323.1		16	33.26	odd	10	inner
363.3.h.l.323.4		16	11.4	even	5	inner
363.3.h.m.245.1		16	33.32	even	2
363.3.h.m.245.4		16	11.10	odd	2
363.3.h.m.251.1		16	11.6	odd	10
363.3.h.m.251.4		16	33.17	even	10
363.3.h.m.269.1		16	33.8	even	10
363.3.h.m.269.4		16	11.8	odd	10
363.3.h.m.323.1		16	11.7	odd	10
363.3.h.m.323.4		16	33.29	even	10
528.3.i.d.353.3		4	132.35	odd	10
528.3.i.d.353.4		4	44.35	even	10