Embedded newform 64.7.j.a.3.17

• Label: 64.7.j.a.3.17
• Level: $64$
• Weight: $7$
• Character: 64.3
• Analytic conductor: $14.723$
• Analytic rank: $0$
• Dimension: $376$
• CM: no
• 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			3.17
Character	\(\chi\)	\(=\)	64.3
Dual form			64.7.j.a.43.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.67029 - 7.10837i) q^{2} +(-35.0021 - 6.96236i) q^{3} +(-37.0579 + 52.1796i) q^{4} +(12.4625 + 8.32721i) q^{5} +(78.9770 + 274.362i) q^{6} +(61.7240 + 149.015i) q^{7} +(506.925 + 71.9073i) q^{8} +(503.167 + 208.419i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 376 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.67029	−	7.10837i	−0.458786	−	0.888547i
							\(3\)	−35.0021	−	6.96236i	−1.29638	−	0.257865i	−0.501775	−	0.864998i	\(-0.667320\pi\)
−0.794600	+	0.607133i	\(0.792320\pi\)
\(5\)	12.4625	+	8.32721i	0.0997004	+	0.0666177i	0.604421	−	0.796665i	\(-0.293404\pi\)
							−0.504721	+	0.863283i	\(0.668404\pi\)
\(7\)	61.7240	+	149.015i	0.179953	+	0.434446i	0.987956	−	0.154733i	\(-0.0494516\pi\)
							−0.808003	+	0.589178i	\(0.799452\pi\)
\(11\)	196.945	+	990.109i	0.147968	+	0.743884i	0.981507	+	0.191424i	\(0.0613106\pi\)
							−0.833540	+	0.552460i	\(0.813689\pi\)
\(13\)	282.982	−	189.083i	0.128804	−	0.0860640i	−0.489499	−	0.872004i	\(-0.662820\pi\)
							0.618303	+	0.785940i	\(0.287820\pi\)
\(17\)	−1417.95	−	1417.95i	−0.288613	−	0.288613i	0.547919	−	0.836532i	\(-0.315420\pi\)
							−0.836532	+	0.547919i	\(0.815420\pi\)
\(19\)	−2460.46	+	1644.02i	−0.358720	+	0.239689i	−0.721850	−	0.692050i	\(-0.756708\pi\)
							0.363130	+	0.931738i	\(0.381708\pi\)
\(23\)	−3913.25	−	1620.92i	−0.321628	−	0.133223i	0.216027	−	0.976387i	\(-0.430690\pi\)
							−0.537656	+	0.843164i	\(0.680690\pi\)
\(29\)	−2000.15	−	397.855i	−0.0820104	−	0.0163129i	0.153915	−	0.988084i	\(-0.450812\pi\)
							−0.235925	+	0.971771i	\(0.575812\pi\)
\(31\)	−2053.02			−0.0689141			−0.0344571	−	0.999406i	\(-0.510970\pi\)
							−0.0344571	+	0.999406i	\(0.510970\pi\)
\(37\)	35621.1	−	53310.8i	0.703238	−	1.05247i	−0.292134	−	0.956377i	\(-0.594365\pi\)
							0.995372	−	0.0960932i	\(-0.0306347\pi\)
\(41\)	43190.8	−	104272.i	0.626671	−	1.51292i	−0.217064	−	0.976157i	\(-0.569648\pi\)
							0.843735	−	0.536760i	\(-0.180352\pi\)
\(43\)	117914.	−	23454.6i	1.48307	−	0.295000i	0.613847	−	0.789425i	\(-0.289621\pi\)
							0.869220	+	0.494425i	\(0.164621\pi\)
\(47\)	−14435.7	+	14435.7i	−0.139042	+	0.139042i	−0.773202	−	0.634160i	\(-0.781346\pi\)
							0.634160	+	0.773202i	\(0.281346\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.7.j.a.3.17	✓	376
64.43	odd	16	inner	64.7.j.a.43.17	yes	376

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.7.j.a.3.17	✓	376	1.1	even	1	trivial
64.7.j.a.43.17	yes	376	64.43	odd	16	inner