Embedded newform 256.2.m.a.101.23

• Label: 256.2.m.a.101.23
• Level: $256$
• Weight: $2$
• Character: 256.101
• Analytic conductor: $2.044$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	256.m (of order \(64\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.04417029174\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(31\) over \(\Q(\zeta_{64})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{64}]$

Embedding invariants

Embedding label			101.23
Character	\(\chi\)	\(=\)	256.101
Dual form			256.2.m.a.109.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.782606 + 1.17793i) q^{2} +(0.703766 - 0.176284i) q^{3} +(-0.775056 + 1.84372i) q^{4} +(1.02351 + 0.484084i) q^{5} +(0.758422 + 0.691029i) q^{6} +(0.442787 + 1.45967i) q^{7} +(-2.77834 + 0.529938i) q^{8} +(-2.18155 + 1.16606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 32 q^{2} - 32 q^{3} - 32 q^{4} - 32 q^{5} - 32 q^{6} - 32 q^{7} - 32 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(255\)
\(\chi(n)\)	\(e\left(\frac{9}{64}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.782606	+	1.17793i	0.553386	+	0.832925i
							\(3\)	0.703766	−	0.176284i	0.406319	−	0.101778i	−0.0344317	−	0.999407i	\(-0.510962\pi\)
0.440751	+	0.897629i	\(0.354712\pi\)
\(5\)	1.02351	+	0.484084i	0.457727	+	0.216489i	0.642490	−	0.766294i	\(-0.277902\pi\)
							−0.184762	+	0.982783i	\(0.559152\pi\)
\(7\)	0.442787	+	1.45967i	0.167358	+	0.551704i	0.999998	−	0.00211032i	\(-0.000671737\pi\)
							−0.832640	+	0.553814i	\(0.813172\pi\)
\(11\)	1.21742	−	0.180588i	0.367067	−	0.0544493i	0.0393509	−	0.999225i	\(-0.487471\pi\)
							0.327716	+	0.944776i	\(0.393721\pi\)
\(13\)	1.16741	−	3.26269i	0.323781	−	0.904907i	−0.663816	−	0.747896i	\(-0.731064\pi\)
							0.987597	−	0.157011i	\(-0.0501858\pi\)
\(17\)	4.99019	−	0.992610i	1.21030	−	0.240743i	0.451621	−	0.892210i	\(-0.350846\pi\)
							0.758677	+	0.651467i	\(0.225846\pi\)
\(19\)	2.64376	+	2.39616i	0.606519	+	0.549717i	0.915995	−	0.401191i	\(-0.131403\pi\)
							−0.309475	+	0.950907i	\(0.600153\pi\)
\(23\)	−3.37113	+	0.332027i	−0.702929	+	0.0692324i	−0.443165	−	0.896440i	\(-0.646145\pi\)
							−0.259763	+	0.965672i	\(0.583645\pi\)
\(29\)	−1.72897	−	2.33124i	−0.321061	−	0.432901i	0.612963	−	0.790112i	\(-0.289978\pi\)
							−0.934024	+	0.357211i	\(0.883728\pi\)
\(31\)	−0.335706	−	0.139054i	−0.0602947	−	0.0249749i	0.352332	−	0.935875i	\(-0.385389\pi\)
							−0.412627	+	0.910900i	\(0.635389\pi\)
\(37\)	0.167221	−	3.40385i	0.0274909	−	0.559590i	−0.945060	−	0.326898i	\(-0.893997\pi\)
							0.972551	−	0.232692i	\(-0.0747534\pi\)
\(41\)	6.53831	−	7.96695i	1.02111	−	1.24423i	0.0517322	−	0.998661i	\(-0.483526\pi\)
							0.969380	−	0.245567i	\(-0.0789742\pi\)
\(43\)	0.966750	−	3.85948i	0.147428	−	0.588566i	−0.850601	−	0.525812i	\(-0.823762\pi\)
							0.998029	−	0.0627539i	\(-0.0199883\pi\)
\(47\)	−4.45634	−	2.97763i	−0.650024	−	0.434332i	0.186356	−	0.982482i	\(-0.440332\pi\)
							−0.836380	+	0.548150i	\(0.815332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.2.m.a.101.23	✓	992
256.109	even	64	inner	256.2.m.a.109.23	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.m.a.101.23	✓	992	1.1	even	1	trivial
256.2.m.a.109.23	yes	992	256.109	even	64	inner