Embedded newform 128.4.k.a.5.15

• Label: 128.4.k.a.5.15
• Level: $128$
• Weight: $4$
• Character: 128.5
• Analytic conductor: $7.552$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.k (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.55224448073\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(47\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			5.15
Character	\(\chi\)	\(=\)	128.5
Dual form			128.4.k.a.77.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.79489 - 2.18595i) q^{2} +(2.18691 + 0.663393i) q^{3} +(-1.55674 + 7.84707i) q^{4} +(-5.13671 - 0.505922i) q^{5} +(-2.47513 - 5.97120i) q^{6} +(4.12327 + 6.17091i) q^{7} +(19.9475 - 10.6817i) q^{8} +(-18.1072 - 12.0988i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 752 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.79489	−	2.18595i	−0.634589	−	0.772849i
							\(3\)	2.18691	+	0.663393i	0.420872	+	0.127670i	0.493615	−	0.869681i	\(-0.335675\pi\)
−0.0727430	+	0.997351i	\(0.523175\pi\)
\(5\)	−5.13671	−	0.505922i	−0.459442	−	0.0452510i	−0.134351	−	0.990934i	\(-0.542895\pi\)
							−0.325091	+	0.945683i	\(0.605395\pi\)
\(7\)	4.12327	+	6.17091i	0.222636	+	0.333198i	0.925926	−	0.377705i	\(-0.123287\pi\)
							−0.703290	+	0.710903i	\(0.748287\pi\)
\(11\)	15.6396	−	29.2596i	0.428683	−	0.802009i	−0.571140	−	0.820853i	\(-0.693498\pi\)
							0.999822	+	0.0188437i	\(0.00599848\pi\)
\(13\)	−6.22716	−	63.2254i	−0.132854	−	1.34889i	−0.798217	−	0.602370i	\(-0.794223\pi\)
							0.665363	−	0.746520i	\(-0.268277\pi\)
\(17\)	58.8385	−	24.3717i	0.839437	−	0.347706i	0.0788052	−	0.996890i	\(-0.474889\pi\)
							0.760631	+	0.649184i	\(0.224889\pi\)
\(19\)	20.0877	−	24.4769i	0.242549	−	0.295547i	−0.637480	−	0.770467i	\(-0.720023\pi\)
							0.880029	+	0.474920i	\(0.157523\pi\)
\(23\)	−10.0307	−	50.4279i	−0.0909370	−	0.457171i	−0.999244	−	0.0388679i	\(-0.987625\pi\)
							0.908307	−	0.418304i	\(-0.137375\pi\)
\(29\)	−28.3738	+	15.1661i	−0.181685	+	0.0971128i	−0.559732	−	0.828674i	\(-0.689096\pi\)
							0.378047	+	0.925787i	\(0.376596\pi\)
\(31\)	−114.348	−	114.348i	−0.662499	−	0.662499i	0.293469	−	0.955968i	\(-0.405190\pi\)
							−0.955968	+	0.293469i	\(0.905190\pi\)
\(37\)	−70.5865	+	57.9289i	−0.313631	+	0.257391i	−0.778018	−	0.628242i	\(-0.783775\pi\)
							0.464387	+	0.885633i	\(0.346275\pi\)
\(41\)	101.472	−	20.1841i	0.386519	−	0.0768834i	0.00199408	−	0.999998i	\(-0.499365\pi\)
							0.384525	+	0.923115i	\(0.374365\pi\)
\(43\)	117.546	−	35.6572i	0.416874	−	0.126457i	−0.0748778	−	0.997193i	\(-0.523857\pi\)
							0.491752	+	0.870735i	\(0.336357\pi\)
\(47\)	62.4139	+	150.681i	0.193702	+	0.467639i	0.990653	−	0.136406i	\(-0.0435550\pi\)
							−0.796951	+	0.604044i	\(0.793555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.k.a.5.15	✓	752
128.77	even	32	inner	128.4.k.a.77.15	yes	752

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.k.a.5.15	✓	752	1.1	even	1	trivial
128.4.k.a.77.15	yes	752	128.77	even	32	inner