Embedded newform 64.12.i.a.5.15

• Label: 64.12.i.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	64.5
Dual form			64.12.i.a.13.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-38.5940 + 23.6326i) q^{2} +(341.547 + 228.214i) q^{3} +(930.998 - 1824.16i) q^{4} +(6128.98 + 1219.13i) q^{5} +(-18575.0 - 736.060i) q^{6} +(-12167.5 + 29374.9i) q^{7} +(7178.65 + 92403.5i) q^{8} +(-3218.78 - 7770.81i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 696 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−38.5940	+	23.6326i	−0.852816	+	0.522212i
							\(3\)	341.547	+	228.214i	0.811490	+	0.542220i	0.890676	−	0.454640i	\(-0.150232\pi\)
−0.0791854	+	0.996860i	\(0.525232\pi\)
\(5\)	6128.98	+	1219.13i	0.877108	+	0.174468i	0.613055	−	0.790040i	\(-0.289940\pi\)
							0.264053	+	0.964508i	\(0.414940\pi\)
\(7\)	−12167.5	+	29374.9i	−0.273628	+	0.660597i	−0.999633	−	0.0270929i	\(-0.991375\pi\)
							0.726005	+	0.687690i	\(0.241375\pi\)
\(11\)	−100929.	−	151051.i	−0.188954	−	0.282789i	0.724882	−	0.688873i	\(-0.241894\pi\)
							−0.913836	+	0.406084i	\(0.866894\pi\)
\(13\)	−1.13564e6	+	225892.i	−0.848302	+	0.168738i	−0.600055	−	0.799959i	\(-0.704855\pi\)
							−0.248247	+	0.968697i	\(0.579855\pi\)
\(17\)	3.75060e6	−	3.75060e6i	0.640666	−	0.640666i	−0.310053	−	0.950719i	\(-0.600347\pi\)
							0.950719	+	0.310053i	\(0.100347\pi\)
\(19\)	−2.94254e6	−	1.47932e7i	−0.272633	−	1.37062i	−0.837952	−	0.545745i	\(-0.816247\pi\)
							0.565319	−	0.824872i	\(-0.308753\pi\)
\(23\)	−5.66555e7	+	2.34675e7i	−1.83544	+	0.760262i	−0.873593	+	0.486657i	\(0.838216\pi\)
							−0.961842	+	0.273605i	\(0.911784\pi\)
\(29\)	−1.00013e8	+	1.49679e8i	−0.905452	+	1.35511i	0.0292117	+	0.999573i	\(0.490700\pi\)
							−0.934664	+	0.355532i	\(0.884300\pi\)
\(31\)			2.64199e8i			1.65746i	0.559650	+	0.828729i	\(0.310936\pi\)
							−0.559650	+	0.828729i	\(0.689064\pi\)
\(37\)	1.41262e8	−	7.10174e8i	0.334902	−	1.68366i	−0.335797	−	0.941934i	\(-0.609006\pi\)
							0.670698	−	0.741730i	\(-0.265994\pi\)
\(41\)	−5.51534e8	+	2.28453e8i	−0.743466	+	0.307954i	−0.722072	−	0.691818i	\(-0.756810\pi\)
							−0.0213933	+	0.999771i	\(0.506810\pi\)
\(43\)	−1.74946e7	+	1.16895e7i	−0.0181479	+	0.0121261i	−0.564611	−	0.825357i	\(-0.690974\pi\)
							0.546463	+	0.837483i	\(0.315974\pi\)
\(47\)	−1.33989e9	+	1.33989e9i	−0.852179	+	0.852179i	−0.990401	−	0.138223i	\(-0.955861\pi\)
							0.138223	+	0.990401i	\(0.455861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.i.a.5.15	✓	696
64.13	even	16	inner	64.12.i.a.13.15	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.12.i.a.5.15	✓	696	1.1	even	1	trivial
64.12.i.a.13.15	yes	696	64.13	even	16	inner