Embedded newform 64.5.j.a.43.3

• Label: 64.5.j.a.43.3
• Level: $64$
• Weight: $5$
• Character: 64.43
• Analytic conductor: $6.616$
• Analytic rank: $0$
• Dimension: $248$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	64.j (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.61567763737\)
Analytic rank:	\(0\)
Dimension:	\(248\)
Relative dimension:	\(31\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			43.3
Character	\(\chi\)	\(=\)	64.43
Dual form			64.5.j.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.78468 - 1.29468i) q^{2} +(4.98643 - 0.991863i) q^{3} +(12.6476 + 9.79988i) q^{4} +(-24.9007 + 16.6381i) q^{5} +(-20.1562 - 2.70193i) q^{6} +(16.4913 - 39.8136i) q^{7} +(-35.1796 - 53.4640i) q^{8} +(-50.9535 + 21.1056i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 248 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(63\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.78468	−	1.29468i	−0.946170	−	0.323669i
							\(3\)	4.98643	−	0.991863i	0.554048	−	0.110207i	0.0898775	−	0.995953i	\(-0.471352\pi\)
0.464171	+	0.885746i	\(0.346352\pi\)
\(5\)	−24.9007	+	16.6381i	−0.996027	+	0.665524i	−0.942904	−	0.333064i	\(-0.891918\pi\)
							−0.0531230	+	0.998588i	\(0.516918\pi\)
\(7\)	16.4913	−	39.8136i	0.336558	−	0.812522i	−0.661484	−	0.749960i	\(-0.730073\pi\)
							0.998041	−	0.0625621i	\(-0.0199272\pi\)
\(11\)	−1.15335	+	5.79830i	−0.00953185	+	0.0479199i	−0.985258	−	0.171072i	\(-0.945277\pi\)
							0.975727	+	0.218992i	\(0.0702769\pi\)
\(13\)	−142.622	−	95.2968i	−0.843915	−	0.563886i	0.0567557	−	0.998388i	\(-0.481924\pi\)
							−0.900671	+	0.434502i	\(0.856924\pi\)
\(17\)	−357.469	+	357.469i	−1.23692	+	1.23692i	−0.275663	+	0.961254i	\(0.588897\pi\)
							−0.961254	+	0.275663i	\(0.911103\pi\)
\(19\)	−336.361	−	224.749i	−0.931747	−	0.622573i	−0.00570056	−	0.999984i	\(-0.501815\pi\)
							−0.926046	+	0.377411i	\(0.876815\pi\)
\(23\)	−480.396	+	198.987i	−0.908122	+	0.376156i	−0.787337	−	0.616523i	\(-0.788541\pi\)
							−0.120784	+	0.992679i	\(0.538541\pi\)
\(29\)	892.973	−	177.623i	1.06180	−	0.211205i	0.366862	−	0.930275i	\(-0.380432\pi\)
							0.694937	+	0.719071i	\(0.255432\pi\)
\(31\)	1476.43			1.53635			0.768174	−	0.640242i	\(-0.221166\pi\)
							0.768174	+	0.640242i	\(0.221166\pi\)
\(37\)	−735.800	−	1101.20i	−0.537472	−	0.804384i	0.458989	−	0.888442i	\(-0.348212\pi\)
							−0.996461	+	0.0840581i	\(0.973212\pi\)
\(41\)	−616.031	−	1487.23i	−0.366467	−	0.884730i	−0.994323	−	0.106401i	\(-0.966067\pi\)
							0.627856	−	0.778329i	\(-0.283933\pi\)
\(43\)	−2111.94	−	420.090i	−1.14221	−	0.227199i	−0.412498	−	0.910959i	\(-0.635344\pi\)
							−0.729707	+	0.683760i	\(0.760344\pi\)
\(47\)	375.581	+	375.581i	0.170023	+	0.170023i	0.786990	−	0.616966i	\(-0.211639\pi\)
							−0.616966	+	0.786990i	\(0.711639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.5.j.a.43.3	yes	248
64.3	odd	16	inner	64.5.j.a.3.3	✓	248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.5.j.a.3.3	✓	248	64.3	odd	16	inner
64.5.j.a.43.3	yes	248	1.1	even	1	trivial