Embedded newform 484.2.g.f.239.1

• Label: 484.2.g.f.239.1
• Level: $484$
• Weight: $2$
• Character: 484.239
• Analytic conductor: $3.865$
• Analytic rank: $0$
• Dimension: $16$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 484 = 2^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	484.g (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.86475945783\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 5*x^15 + 13*x^14 - 25*x^13 + 35*x^12 - 30*x^11 - 2*x^10 + 60*x^9 - 116*x^8 + 120*x^7 - 8*x^6 - 240*x^5 + 560*x^4 - 800*x^3 + 832*x^2 - 640*x + 256
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			239.1
Root			\(-0.204982 + 1.39928i\) of defining polynomial
Character	\(\chi\)	\(=\)	484.239
Dual form			484.2.g.f.403.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26745 + 0.627351i) q^{2} +(0.873506 - 0.283819i) q^{3} +(1.21286 - 1.59027i) q^{4} +(-1.38723 - 1.00788i) q^{5} +(-0.929072 + 0.907722i) q^{6} +(-0.738720 + 2.27355i) q^{7} +(-0.539585 + 2.77648i) q^{8} +(-1.74459 + 1.26752i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 5 q^{2} - q^{4} + 4 q^{5} + 10 q^{6} - 5 q^{8} + 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\)	\(243\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26745	+	0.627351i	−0.896223	+	0.443604i
							\(3\)	0.873506	−	0.283819i	0.504319	−	0.163863i	−0.0457980	−	0.998951i	\(-0.514583\pi\)
0.550117	+	0.835088i	\(0.314583\pi\)
\(5\)	−1.38723	−	1.00788i	−0.620390	−	0.450740i	0.232668	−	0.972556i	\(-0.425254\pi\)
							−0.853058	+	0.521817i	\(0.825254\pi\)
\(7\)	−0.738720	+	2.27355i	−0.279210	+	0.859319i	0.708865	+	0.705344i	\(0.249208\pi\)
							−0.988075	+	0.153975i	\(0.950792\pi\)
\(11\)		0			0
							\(13\)	−0.945515	−	1.30139i	−0.262239	−	0.360941i	0.657512	−	0.753444i	\(-0.271609\pi\)
−0.919750	+	0.392504i	\(0.871609\pi\)
\(17\)	−3.55361	+	4.89112i	−0.861877	+	1.18627i	0.119242	+	0.992865i	\(0.461954\pi\)
							−0.981119	+	0.193406i	\(0.938046\pi\)
\(19\)	−0.409609	−	1.26065i	−0.0939707	−	0.289212i	0.893013	−	0.450030i	\(-0.148587\pi\)
							−0.986984	+	0.160818i	\(0.948587\pi\)
\(23\)			7.25726i			1.51324i	0.653853	+	0.756622i	\(0.273152\pi\)
							−0.653853	+	0.756622i	\(0.726848\pi\)
\(29\)	0.516085	+	0.167686i	0.0958346	+	0.0311386i	0.356542	−	0.934279i	\(-0.383956\pi\)
							−0.260707	+	0.965418i	\(0.583956\pi\)
\(31\)	4.83783	+	6.65870i	0.868899	+	1.19594i	0.979373	+	0.202058i	\(0.0647631\pi\)
							−0.110474	+	0.993879i	\(0.535237\pi\)
\(37\)	0.114237	−	0.351585i	0.0187804	−	0.0578002i	−0.941227	−	0.337774i	\(-0.890326\pi\)
							0.960008	+	0.279974i	\(0.0903259\pi\)
\(41\)	−4.84269	+	1.57349i	−0.756301	+	0.245737i	−0.661690	−	0.749777i	\(-0.730161\pi\)
							−0.0946108	+	0.995514i	\(0.530161\pi\)
\(43\)	−9.45922			−1.44252			−0.721259	−	0.692666i	\(-0.756436\pi\)
							−0.721259	+	0.692666i	\(0.756436\pi\)
\(47\)	−3.81125	+	1.23835i	−0.555928	+	0.180632i	−0.573488	−	0.819214i	\(-0.694410\pi\)
							0.0175599	+	0.999846i	\(0.494410\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	484.2.g.f.239.1		16
4.3	odd	2	inner	484.2.g.f.239.4		16
11.2	odd	10		484.2.c.d.483.1		16
11.3	even	5		44.2.g.a.39.4	yes	16
11.4	even	5		484.2.g.j.403.1		16
11.5	even	5		484.2.g.i.475.2		16
11.6	odd	10		44.2.g.a.35.3	✓	16
11.7	odd	10	inner	484.2.g.f.403.4		16
11.8	odd	10		484.2.g.i.215.1		16
11.9	even	5		484.2.c.d.483.16		16
11.10	odd	2		484.2.g.j.239.4		16
33.14	odd	10		396.2.r.a.127.1		16
33.17	even	10		396.2.r.a.343.2		16
44.3	odd	10		44.2.g.a.39.3	yes	16
44.7	even	10	inner	484.2.g.f.403.1		16
44.15	odd	10		484.2.g.j.403.4		16
44.19	even	10		484.2.g.i.215.2		16
44.27	odd	10		484.2.g.i.475.1		16
44.31	odd	10		484.2.c.d.483.2		16
44.35	even	10		484.2.c.d.483.15		16
44.39	even	10		44.2.g.a.35.4	yes	16
44.43	even	2		484.2.g.j.239.1		16
88.3	odd	10		704.2.u.c.127.2		16
88.61	odd	10		704.2.u.c.255.2		16
88.69	even	10		704.2.u.c.127.3		16
88.83	even	10		704.2.u.c.255.3		16
132.47	even	10		396.2.r.a.127.2		16
132.83	odd	10		396.2.r.a.343.1		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.g.a.35.3	✓	16	11.6	odd	10
44.2.g.a.35.4	yes	16	44.39	even	10
44.2.g.a.39.3	yes	16	44.3	odd	10
44.2.g.a.39.4	yes	16	11.3	even	5
396.2.r.a.127.1		16	33.14	odd	10
396.2.r.a.127.2		16	132.47	even	10
396.2.r.a.343.1		16	132.83	odd	10
396.2.r.a.343.2		16	33.17	even	10
484.2.c.d.483.1		16	11.2	odd	10
484.2.c.d.483.2		16	44.31	odd	10
484.2.c.d.483.15		16	44.35	even	10
484.2.c.d.483.16		16	11.9	even	5
484.2.g.f.239.1		16	1.1	even	1	trivial
484.2.g.f.239.4		16	4.3	odd	2	inner
484.2.g.f.403.1		16	44.7	even	10	inner
484.2.g.f.403.4		16	11.7	odd	10	inner
484.2.g.i.215.1		16	11.8	odd	10
484.2.g.i.215.2		16	44.19	even	10
484.2.g.i.475.1		16	44.27	odd	10
484.2.g.i.475.2		16	11.5	even	5
484.2.g.j.239.1		16	44.43	even	2
484.2.g.j.239.4		16	11.10	odd	2
484.2.g.j.403.1		16	11.4	even	5
484.2.g.j.403.4		16	44.15	odd	10
704.2.u.c.127.2		16	88.3	odd	10
704.2.u.c.127.3		16	88.69	even	10
704.2.u.c.255.2		16	88.61	odd	10
704.2.u.c.255.3		16	88.83	even	10