Embedded newform 128.5.l.a.3.1

• Label: 128.5.l.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.98767 + 0.313821i) q^{2} +(-13.0973 + 10.7487i) q^{3} +(15.8030 - 2.50283i) q^{4} +(37.0978 + 11.2535i) q^{5} +(48.8546 - 46.9724i) q^{6} +(4.64899 + 23.3721i) q^{7} +(-62.2319 + 14.9398i) q^{8} +(40.2030 - 202.114i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.98767	+	0.313821i	−0.996918	+	0.0784552i
							\(3\)	−13.0973	+	10.7487i	−1.45526	+	1.19430i	−0.513460	+	0.858113i	\(0.671637\pi\)
−0.941796	+	0.336185i	\(0.890863\pi\)
\(5\)	37.0978	+	11.2535i	1.48391	+	0.450140i	0.925283	−	0.379277i	\(-0.123827\pi\)
							0.558630	+	0.829417i	\(0.311327\pi\)
\(7\)	4.64899	+	23.3721i	0.0948774	+	0.476981i	0.998786	+	0.0492557i	\(0.0156849\pi\)
							−0.903909	+	0.427725i	\(0.859315\pi\)
\(11\)	−7.43512	−	75.4900i	−0.0614473	−	0.623885i	−0.975755	−	0.218865i	\(-0.929765\pi\)
							0.914308	−	0.405020i	\(-0.132735\pi\)
\(13\)	−37.0665	−	122.192i	−0.219328	−	0.723029i	−0.995690	−	0.0927413i	\(-0.970437\pi\)
							0.776362	−	0.630287i	\(-0.217063\pi\)
\(17\)	106.184	−	256.351i	0.367420	−	0.887029i	−0.626752	−	0.779219i	\(-0.715616\pi\)
							0.994171	−	0.107810i	\(-0.0343840\pi\)
\(19\)	231.466	−	433.042i	0.641180	−	1.19956i	−0.325976	−	0.945378i	\(-0.605693\pi\)
							0.967156	−	0.254185i	\(-0.0818071\pi\)
\(23\)	−5.19074	+	3.46834i	−0.00981237	+	0.00655642i	−0.560467	−	0.828177i	\(-0.689378\pi\)
							0.550654	+	0.834733i	\(0.314378\pi\)
\(29\)	132.905	−	1349.41i	0.158032	−	1.60453i	−0.509853	−	0.860262i	\(-0.670300\pi\)
							0.667885	−	0.744265i	\(-0.267200\pi\)
\(31\)	689.095	+	689.095i	0.717060	+	0.717060i	0.968002	−	0.250942i	\(-0.0807403\pi\)
							−0.250942	+	0.968002i	\(0.580740\pi\)
\(37\)	−405.257	−	758.183i	−0.296024	−	0.553822i	0.689308	−	0.724468i	\(-0.257915\pi\)
							−0.985332	+	0.170646i	\(0.945415\pi\)
\(41\)	−52.1466	+	34.8432i	−0.0310212	+	0.0207277i	−0.570984	−	0.820961i	\(-0.693438\pi\)
							0.539963	+	0.841689i	\(0.318438\pi\)
\(43\)	−2711.28	−	2225.09i	−1.46635	−	1.20340i	−0.934199	−	0.356752i	\(-0.883884\pi\)
							−0.532150	−	0.846650i	\(-0.678616\pi\)
\(47\)	69.2697	−	167.232i	0.0313579	−	0.0757047i	−0.907425	−	0.420214i	\(-0.861955\pi\)
							0.938783	+	0.344510i	\(0.111955\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.1	✓	1008
128.43	odd	32	inner	128.5.l.a.43.1	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.1	✓	1008	1.1	even	1	trivial
128.5.l.a.43.1	yes	1008	128.43	odd	32	inner