Embedded newform 64.7.j.a.3.4

• Label: 64.7.j.a.3.4
• Level: $64$
• Weight: $7$
• Character: 64.3
• Analytic conductor: $14.723$
• Analytic rank: $0$
• Dimension: $376$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	64.j (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.7234613517\)
Analytic rank:	\(0\)
Dimension:	\(376\)
Relative dimension:	\(47\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	64.3
Dual form			64.7.j.a.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.86826 + 1.44584i) q^{2} +(-21.9073 - 4.35764i) q^{3} +(59.8191 - 22.7524i) q^{4} +(90.4060 + 60.4074i) q^{5} +(178.673 + 2.61266i) q^{6} +(-110.859 - 267.638i) q^{7} +(-437.776 + 265.511i) q^{8} +(-212.566 - 88.0477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 376 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(63\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.86826	+	1.44584i	−0.983533	+	0.180729i
							\(3\)	−21.9073	−	4.35764i	−0.811383	−	0.161394i	−0.228073	−	0.973644i	\(-0.573243\pi\)
−0.583310	+	0.812250i	\(0.698243\pi\)
\(5\)	90.4060	+	60.4074i	0.723248	+	0.483259i	0.861898	−	0.507082i	\(-0.169276\pi\)
							−0.138649	+	0.990342i	\(0.544276\pi\)
\(7\)	−110.859	−	267.638i	−0.323205	−	0.780286i	−0.999064	−	0.0432553i	\(-0.986227\pi\)
							0.675859	−	0.737031i	\(-0.263773\pi\)
\(11\)	421.666	+	2119.86i	0.316804	+	1.59268i	0.730923	+	0.682460i	\(0.239090\pi\)
							−0.414119	+	0.910223i	\(0.635910\pi\)
\(13\)	−1397.51	+	933.786i	−0.636099	+	0.425028i	−0.831368	−	0.555722i	\(-0.812442\pi\)
							0.195269	+	0.980750i	\(0.437442\pi\)
\(17\)	−4681.37	−	4681.37i	−0.952853	−	0.952853i	0.0460843	−	0.998938i	\(-0.485326\pi\)
							−0.998938	+	0.0460843i	\(0.985326\pi\)
\(19\)	10846.9	−	7247.69i	1.58142	−	1.05667i	0.618892	−	0.785476i	\(-0.287582\pi\)
							0.962525	−	0.271192i	\(-0.0874180\pi\)
\(23\)	9582.99	+	3969.40i	0.787621	+	0.326243i	0.739987	−	0.672621i	\(-0.234832\pi\)
							0.0476345	+	0.998865i	\(0.484832\pi\)
\(29\)	9826.26	+	1954.56i	0.402897	+	0.0801412i	0.392379	−	0.919803i	\(-0.371652\pi\)
							0.0105178	+	0.999945i	\(0.496652\pi\)
\(31\)	17303.4			0.580827			0.290413	−	0.956901i	\(-0.406207\pi\)
							0.290413	+	0.956901i	\(0.406207\pi\)
\(37\)	44623.4	−	66783.7i	0.880963	−	1.31845i	−0.0662348	−	0.997804i	\(-0.521099\pi\)
							0.947198	−	0.320650i	\(-0.103901\pi\)
\(41\)	−1490.51	+	3598.40i	−0.0216263	+	0.0522105i	−0.934324	−	0.356425i	\(-0.883995\pi\)
							0.912697	+	0.408636i	\(0.133995\pi\)
\(43\)	−104101.	+	20706.9i	−1.30933	+	0.260441i	−0.799931	−	0.600092i	\(-0.795131\pi\)
							−0.509394	+	0.860533i	\(0.670131\pi\)
\(47\)	135339.	−	135339.i	1.30355	−	1.30355i	0.377574	−	0.925979i	\(-0.376758\pi\)
							0.925979	−	0.377574i	\(-0.123242\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.7.j.a.3.4	✓	376
64.43	odd	16	inner	64.7.j.a.43.4	yes	376

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.7.j.a.3.4	✓	376	1.1	even	1	trivial
64.7.j.a.43.4	yes	376	64.43	odd	16	inner