Embedded newform 64.12.i.a.5.1

• Label: 64.12.i.a.5.1
• Level: $64$
• Weight: $12$
• Character: 64.5
• Analytic conductor: $49.174$
• Analytic rank: $0$
• Dimension: $696$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	64.i (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(49.1739635558\)
Analytic rank:	\(0\)
Dimension:	\(696\)
Relative dimension:	\(87\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	64.5
Dual form			64.12.i.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-45.2527 + 0.440829i) q^{2} +(491.613 + 328.486i) q^{3} +(2047.61 - 39.8974i) q^{4} +(12129.7 + 2412.76i) q^{5} +(-22391.6 - 14648.1i) q^{6} +(16326.7 - 39416.1i) q^{7} +(-92642.3 + 2708.11i) q^{8} +(65989.7 + 159313. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 696 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−45.2527	+	0.440829i	−0.999953	+	0.00974103i
							\(3\)	491.613	+	328.486i	1.16804	+	0.780458i	0.979468	−	0.201602i	\(-0.0646148\pi\)
0.188570	+	0.982060i	\(0.439615\pi\)
\(5\)	12129.7	+	2412.76i	1.73587	+	0.345286i	0.958789	−	0.284118i	\(-0.0917008\pi\)
							0.777078	+	0.629404i	\(0.216701\pi\)
\(7\)	16326.7	−	39416.1i	0.367162	−	0.886408i	−0.627050	−	0.778979i	\(-0.715738\pi\)
							0.994213	−	0.107430i	\(-0.0342621\pi\)
\(11\)	91469.3	+	136893.i	0.171244	+	0.256285i	0.907158	−	0.420790i	\(-0.138247\pi\)
							−0.735914	+	0.677075i	\(0.763247\pi\)
\(13\)	352949.	−	70206.0i	0.263648	−	0.0524428i	−0.0614964	−	0.998107i	\(-0.519587\pi\)
							0.325144	+	0.945665i	\(0.394587\pi\)
\(17\)	−1.23840e6	+	1.23840e6i	−0.211540	+	0.211540i	−0.804922	−	0.593381i	\(-0.797793\pi\)
							0.593381	+	0.804922i	\(0.297793\pi\)
\(19\)	2.64003e6	+	1.32723e7i	0.244604	+	1.22971i	0.886432	+	0.462858i	\(0.153176\pi\)
							−0.641828	+	0.766848i	\(0.721824\pi\)
\(23\)	−3.11778e6	+	1.29143e6i	−0.101005	+	0.0418376i	−0.432614	−	0.901579i	\(-0.642409\pi\)
							0.331609	+	0.943417i	\(0.392409\pi\)
\(29\)	−1.16467e8	+	1.74305e8i	−1.05442	+	1.57805i	−0.264942	+	0.964264i	\(0.585353\pi\)
							−0.789475	+	0.613782i	\(0.789647\pi\)
\(31\)		−	1.45051e8i		−	0.909981i	−0.890496	−	0.454991i	\(-0.849643\pi\)
							0.890496	−	0.454991i	\(-0.150357\pi\)
\(37\)	−7.97015e6	+	4.00687e7i	−0.0188955	+	0.0949938i	−0.989082	−	0.147365i	\(-0.952921\pi\)
							0.970187	+	0.242358i	\(0.0779210\pi\)
\(41\)	1.04333e9	−	4.32163e8i	1.40641	−	0.582555i	0.455004	−	0.890489i	\(-0.349638\pi\)
							0.951408	+	0.307934i	\(0.0996377\pi\)
\(43\)	−4.52598e7	+	3.02416e7i	−0.0469501	+	0.0313710i	−0.578824	−	0.815453i	\(-0.696488\pi\)
							0.531874	+	0.846824i	\(0.321488\pi\)
\(47\)	2.83350e8	−	2.83350e8i	0.180213	−	0.180213i	−0.611236	−	0.791449i	\(-0.709327\pi\)
							0.791449	+	0.611236i	\(0.209327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.i.a.5.1	✓	696
64.13	even	16	inner	64.12.i.a.13.1	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.12.i.a.5.1	✓	696	1.1	even	1	trivial
64.12.i.a.13.1	yes	696	64.13	even	16	inner