Embedded newform 128.5.l.a.3.18

• Label: 128.5.l.a.3.18
• Level: $128$
• Weight: $5$
• Character: 128.3
• Analytic conductor: $13.231$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2313552747\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(63\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.18
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66833 - 2.97993i) q^{2} +(8.56994 - 7.03317i) q^{3} +(-1.75999 + 15.9029i) q^{4} +(16.1113 + 4.88731i) q^{5} +(-43.8258 - 6.77100i) q^{6} +(8.61826 + 43.3269i) q^{7} +(52.0858 - 37.1896i) q^{8} +(8.17609 - 41.1040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66833	−	2.97993i	−0.667083	−	0.744983i
							\(3\)	8.56994	−	7.03317i	0.952215	−	0.781463i	−0.0236760	−	0.999720i	\(-0.507537\pi\)
0.975891	+	0.218257i	\(0.0700370\pi\)
\(5\)	16.1113	+	4.88731i	0.644452	+	0.195492i	0.595559	−	0.803312i	\(-0.296931\pi\)
							0.0488933	+	0.998804i	\(0.484431\pi\)
\(7\)	8.61826	+	43.3269i	0.175883	+	0.884223i	0.963428	+	0.267966i	\(0.0863516\pi\)
							−0.787545	+	0.616257i	\(0.788648\pi\)
\(11\)	9.40173	+	95.4574i	0.0777003	+	0.788904i	0.952524	+	0.304463i	\(0.0984770\pi\)
							−0.874824	+	0.484441i	\(0.839023\pi\)
\(13\)	85.0676	+	280.430i	0.503359	+	1.65935i	0.728516	+	0.685029i	\(0.240210\pi\)
							−0.225158	+	0.974322i	\(0.572290\pi\)
\(17\)	−98.7544	+	238.414i	−0.341711	+	0.824962i	0.655832	+	0.754907i	\(0.272318\pi\)
							−0.997543	+	0.0700560i	\(0.977682\pi\)
\(19\)	−116.932	+	218.764i	−0.323911	+	0.605994i	−0.990193	−	0.139705i	\(-0.955385\pi\)
							0.666282	+	0.745700i	\(0.267885\pi\)
\(23\)	91.3375	−	61.0297i	0.172661	−	0.115368i	−0.466237	−	0.884660i	\(-0.654391\pi\)
							0.638898	+	0.769292i	\(0.279391\pi\)
\(29\)	134.588	−	1366.49i	0.160033	−	1.62485i	−0.494755	−	0.869033i	\(-0.664742\pi\)
							0.654788	−	0.755813i	\(-0.272758\pi\)
\(31\)	165.272	+	165.272i	0.171979	+	0.171979i	0.787848	−	0.615869i	\(-0.211195\pi\)
							−0.615869	+	0.787848i	\(0.711195\pi\)
\(37\)	−616.425	−	1153.25i	−0.450274	−	0.842403i	−0.999975	−	0.00704161i	\(-0.997759\pi\)
							0.549702	−	0.835361i	\(-0.314741\pi\)
\(41\)	2025.91	−	1353.67i	1.20518	−	0.805276i	0.219784	−	0.975549i	\(-0.429465\pi\)
							0.985397	+	0.170272i	\(0.0544648\pi\)
\(43\)	−356.343	−	292.443i	−0.192722	−	0.158163i	0.533084	−	0.846062i	\(-0.321033\pi\)
							−0.725806	+	0.687899i	\(0.758533\pi\)
\(47\)	1147.58	−	2770.51i	0.519503	−	1.25419i	−0.418706	−	0.908122i	\(-0.637516\pi\)
							0.938209	−	0.346070i	\(-0.112484\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.18	✓	1008
128.43	odd	32	inner	128.5.l.a.43.18	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.18	✓	1008	1.1	even	1	trivial
128.5.l.a.43.18	yes	1008	128.43	odd	32	inner