Embedded newform 128.5.l.a.3.17

• Label: 128.5.l.a.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.80758 + 2.84912i) q^{2} +(-0.236785 + 0.194325i) q^{3} +(-0.235026 - 15.9983i) q^{4} +(-27.1487 - 8.23546i) q^{5} +(0.111138 - 1.22021i) q^{6} +(13.1413 + 66.0660i) q^{7} +(46.2409 + 44.2468i) q^{8} +(-15.7840 + 79.3516i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.80758	+	2.84912i	−0.701894	+	0.712281i
							\(3\)	−0.236785	+	0.194325i	−0.0263095	+	0.0215916i	−0.647456	−	0.762103i	\(-0.724167\pi\)
0.621147	+	0.783694i	\(0.286667\pi\)
\(5\)	−27.1487	−	8.23546i	−1.08595	−	0.329418i	−0.303942	−	0.952691i	\(-0.598303\pi\)
							−0.782005	+	0.623272i	\(0.785803\pi\)
\(7\)	13.1413	+	66.0660i	0.268191	+	1.34829i	0.846466	+	0.532444i	\(0.178726\pi\)
							−0.578275	+	0.815842i	\(0.696274\pi\)
\(11\)	9.35266	+	94.9592i	0.0772947	+	0.784786i	0.953207	+	0.302317i	\(0.0977601\pi\)
							−0.875913	+	0.482470i	\(0.839740\pi\)
\(13\)	−89.6434	−	295.515i	−0.530435	−	1.74861i	−0.654225	−	0.756300i	\(-0.727005\pi\)
							0.123791	−	0.992308i	\(-0.460495\pi\)
\(17\)	84.3133	−	203.550i	0.291742	−	0.704327i	−0.708257	−	0.705955i	\(-0.750518\pi\)
							0.999999	+	0.00162800i	\(0.000518209\pi\)
\(19\)	240.129	−	449.250i	0.665178	−	1.24446i	−0.292403	−	0.956295i	\(-0.594455\pi\)
							0.957581	−	0.288165i	\(-0.0930451\pi\)
\(23\)	−347.658	+	232.298i	−0.657198	+	0.439126i	−0.838943	−	0.544220i	\(-0.816826\pi\)
							0.181744	+	0.983346i	\(0.441826\pi\)
\(29\)	88.1839	−	895.347i	0.104856	−	1.06462i	−0.788427	−	0.615128i	\(-0.789104\pi\)
							0.893284	−	0.449494i	\(-0.148396\pi\)
\(31\)	−146.592	−	146.592i	−0.152541	−	0.152541i	0.626711	−	0.779252i	\(-0.284401\pi\)
							−0.779252	+	0.626711i	\(0.784401\pi\)
\(37\)	−495.611	−	927.222i	−0.362024	−	0.677299i	0.633299	−	0.773907i	\(-0.281700\pi\)
							−0.995322	+	0.0966087i	\(0.969200\pi\)
\(41\)	−612.344	+	409.155i	−0.364274	+	0.243400i	−0.724214	−	0.689576i	\(-0.757797\pi\)
							0.359940	+	0.932975i	\(0.382797\pi\)
\(43\)	322.570	+	264.726i	0.174456	+	0.143173i	0.717570	−	0.696487i	\(-0.245255\pi\)
							−0.543113	+	0.839660i	\(0.682755\pi\)
\(47\)	1243.11	−	3001.13i	0.562747	−	1.35859i	−0.344814	−	0.938671i	\(-0.612058\pi\)
							0.907561	−	0.419920i	\(-0.137942\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.17	✓	1008
128.43	odd	32	inner	128.5.l.a.43.17	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.17	✓	1008	1.1	even	1	trivial
128.5.l.a.43.17	yes	1008	128.43	odd	32	inner