Embedded newform 128.2.k.a.85.5

• Label: 128.2.k.a.85.5
• Level: $128$
• Weight: $2$
• Character: 128.85
• Analytic conductor: $1.022$
• Analytic rank: $0$
• Dimension: $240$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.02208514587\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(15\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			85.5
Character	\(\chi\)	\(=\)	128.85
Dual form			128.2.k.a.125.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781170 + 1.17889i) q^{2} +(2.14610 - 2.61503i) q^{3} +(-0.779546 - 1.84182i) q^{4} +(-1.87385 + 0.568427i) q^{5} +(1.40636 + 4.57280i) q^{6} +(1.31028 + 0.260631i) q^{7} +(2.78026 + 0.519781i) q^{8} +(-1.64737 - 8.28191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{29}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.781170	+	1.17889i	−0.552371	+	0.833599i
							\(3\)	2.14610	−	2.61503i	1.23905	−	1.50979i	0.453159	−	0.891430i	\(-0.350297\pi\)
0.785894	−	0.618361i	\(-0.212203\pi\)
\(5\)	−1.87385	+	0.568427i	−0.838012	+	0.254208i	−0.679985	−	0.733226i	\(-0.738014\pi\)
							−0.158028	+	0.987435i	\(0.550514\pi\)
\(7\)	1.31028	+	0.260631i	0.495240	+	0.0985094i	0.436390	−	0.899757i	\(-0.356257\pi\)
							0.0588498	+	0.998267i	\(0.481257\pi\)
\(11\)	1.25098	+	0.123211i	0.377185	+	0.0371495i	0.284833	−	0.958577i	\(-0.408062\pi\)
							0.0923513	+	0.995726i	\(0.470562\pi\)
\(13\)	0.385439	−	1.27062i	0.106902	−	0.352408i	−0.887127	−	0.461525i	\(-0.847302\pi\)
							0.994029	+	0.109117i	\(0.0348025\pi\)
\(17\)	2.09998	+	5.06980i	0.509320	+	1.22961i	0.944276	+	0.329156i	\(0.106764\pi\)
							−0.434955	+	0.900452i	\(0.643236\pi\)
\(19\)	0.832861	−	0.445173i	0.191071	−	0.102130i	−0.373087	−	0.927796i	\(-0.621701\pi\)
							0.564158	+	0.825667i	\(0.309201\pi\)
\(23\)	−4.39281	+	6.57431i	−0.915965	+	1.37084i	0.0127000	+	0.999919i	\(0.495957\pi\)
							−0.928665	+	0.370919i	\(0.879043\pi\)
\(29\)	0.394050	+	4.00086i	0.0731733	+	0.742941i	0.959843	+	0.280537i	\(0.0905127\pi\)
							−0.886670	+	0.462403i	\(0.846987\pi\)
\(31\)	3.39435	+	3.39435i	0.609643	+	0.609643i	0.942853	−	0.333210i	\(-0.108132\pi\)
							−0.333210	+	0.942853i	\(0.608132\pi\)
\(37\)	3.30520	−	6.18360i	0.543372	−	1.01658i	−0.449028	−	0.893518i	\(-0.648230\pi\)
							0.992399	−	0.123059i	\(-0.0392705\pi\)
\(41\)	0.369000	+	0.246558i	0.0576282	+	0.0385059i	0.584051	−	0.811717i	\(-0.301467\pi\)
							−0.526422	+	0.850223i	\(0.676467\pi\)
\(43\)	3.34306	+	4.07353i	0.509811	+	0.621207i	0.962769	−	0.270325i	\(-0.0871312\pi\)
							−0.452958	+	0.891532i	\(0.649631\pi\)
\(47\)	−2.55561	+	1.05857i	−0.372774	+	0.154408i	−0.561202	−	0.827679i	\(-0.689661\pi\)
							0.188428	+	0.982087i	\(0.439661\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.85.5	✓	240
4.3	odd	2		512.2.k.a.177.1		240
128.3	odd	32		512.2.k.a.81.1		240
128.125	even	32	inner	128.2.k.a.125.5	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.85.5	✓	240	1.1	even	1	trivial
128.2.k.a.125.5	yes	240	128.125	even	32	inner
512.2.k.a.81.1		240	128.3	odd	32
512.2.k.a.177.1		240	4.3	odd	2