Embedded newform 128.5.l.a.3.2

• Label: 128.5.l.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.95882 - 0.572521i) q^{2} +(-4.36463 + 3.58196i) q^{3} +(15.3444 + 4.53301i) q^{4} +(2.00607 + 0.608536i) q^{5} +(19.3295 - 11.6815i) q^{6} +(-7.49964 - 37.7032i) q^{7} +(-58.1506 - 26.7304i) q^{8} +(-9.58275 + 48.1757i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.95882	−	0.572521i	−0.989704	−	0.143130i
							\(3\)	−4.36463	+	3.58196i	−0.484959	+	0.397996i	−0.844854	−	0.534997i	\(-0.820313\pi\)
0.359895	+	0.932993i	\(0.382813\pi\)
\(5\)	2.00607	+	0.608536i	0.0802430	+	0.0243414i	0.330151	−	0.943928i	\(-0.392900\pi\)
							−0.249908	+	0.968270i	\(0.580400\pi\)
\(7\)	−7.49964	−	37.7032i	−0.153054	−	0.769454i	−0.978705	−	0.205273i	\(-0.934192\pi\)
							0.825651	−	0.564181i	\(-0.190808\pi\)
\(11\)	11.0714	+	112.410i	0.0914994	+	0.929009i	0.925879	+	0.377819i	\(0.123326\pi\)
							−0.834380	+	0.551190i	\(0.814174\pi\)
\(13\)	−52.4777	−	172.996i	−0.310519	−	1.02364i	−0.963822	−	0.266547i	\(-0.914117\pi\)
							0.653303	−	0.757097i	\(-0.273383\pi\)
\(17\)	−92.2442	+	222.697i	−0.319184	+	0.770579i	0.680114	+	0.733107i	\(0.261930\pi\)
							−0.999298	+	0.0374718i	\(0.988070\pi\)
\(19\)	−42.4182	+	79.3590i	−0.117502	+	0.219831i	−0.933796	−	0.357806i	\(-0.883525\pi\)
							0.816294	+	0.577637i	\(0.196025\pi\)
\(23\)	800.966	−	535.189i	1.51411	−	1.01170i	0.527329	−	0.849661i	\(-0.323193\pi\)
							0.986785	−	0.162038i	\(-0.0518066\pi\)
\(29\)	97.7648	−	992.623i	0.116248	−	1.18029i	−0.743201	−	0.669068i	\(-0.766693\pi\)
							0.859449	−	0.511221i	\(-0.170807\pi\)
\(31\)	−318.713	−	318.713i	−0.331648	−	0.331648i	0.521564	−	0.853212i	\(-0.325349\pi\)
							−0.853212	+	0.521564i	\(0.825349\pi\)
\(37\)	−882.938	−	1651.86i	−0.644951	−	1.20662i	−0.965751	−	0.259472i	\(-0.916451\pi\)
							0.320799	−	0.947147i	\(-0.396049\pi\)
\(41\)	2512.35	−	1678.70i	1.49456	−	0.998631i	0.503676	−	0.863893i	\(-0.331981\pi\)
							0.990881	−	0.134738i	\(-0.0430193\pi\)
\(43\)	1700.95	+	1395.93i	0.919927	+	0.754965i	0.969828	−	0.243789i	\(-0.0783905\pi\)
							−0.0499008	+	0.998754i	\(0.515891\pi\)
\(47\)	264.637	−	638.891i	0.119800	−	0.289222i	−0.852592	−	0.522577i	\(-0.824971\pi\)
							0.972391	+	0.233356i	\(0.0749706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.2	✓	1008
128.43	odd	32	inner	128.5.l.a.43.2	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.2	✓	1008	1.1	even	1	trivial
128.5.l.a.43.2	yes	1008	128.43	odd	32	inner