Embedded newform 64.12.i.a.5.4

• Label: 64.12.i.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	64.5
Dual form			64.12.i.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-44.8708 + 5.88307i) q^{2} +(196.675 + 131.414i) q^{3} +(1978.78 - 527.956i) q^{4} +(-9168.52 - 1823.73i) q^{5} +(-9598.10 - 4739.61i) q^{6} +(6070.55 - 14655.6i) q^{7} +(-85683.4 + 35331.1i) q^{8} +(-46379.8 - 111971. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 696 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−44.8708	+	5.88307i	−0.991514	+	0.129999i
							\(3\)	196.675	+	131.414i	0.467286	+	0.312231i	0.766828	−	0.641853i	\(-0.221834\pi\)
−0.299542	+	0.954083i	\(0.596834\pi\)
\(5\)	−9168.52	−	1823.73i	−1.31209	−	0.260991i	−0.511025	−	0.859566i	\(-0.670734\pi\)
							−0.801067	+	0.598574i	\(0.795734\pi\)
\(7\)	6070.55	−	14655.6i	0.136518	−	0.329583i	−0.840805	−	0.541338i	\(-0.817918\pi\)
							0.977323	+	0.211755i	\(0.0679180\pi\)
\(11\)	−533933.	−	799087.i	−0.999602	−	1.49601i	−0.862886	−	0.505399i	\(-0.831345\pi\)
							−0.136716	−	0.990610i	\(-0.543655\pi\)
\(13\)	−380110.	+	75608.5i	−0.283936	+	0.0564784i	−0.335004	−	0.942217i	\(-0.608738\pi\)
							0.0510676	+	0.998695i	\(0.483738\pi\)
\(17\)	−3.99544e6	+	3.99544e6i	−0.682489	+	0.682489i	−0.960560	−	0.278071i	\(-0.910305\pi\)
							0.278071	+	0.960560i	\(0.410305\pi\)
\(19\)	552587.	+	2.77804e6i	0.0511983	+	0.257391i	0.997904	−	0.0647078i	\(-0.0206115\pi\)
							−0.946706	+	0.322099i	\(0.895612\pi\)
\(23\)	1.10224e7	−	4.56563e6i	0.357086	−	0.147910i	−0.196926	−	0.980418i	\(-0.563096\pi\)
							0.554013	+	0.832508i	\(0.313096\pi\)
\(29\)	−4.08375e7	+	6.11177e7i	−0.369718	+	0.553322i	−0.968951	−	0.247253i	\(-0.920472\pi\)
							0.599233	+	0.800575i	\(0.295472\pi\)
\(31\)		−	1.21450e8i		−	0.761918i	−0.924592	−	0.380959i	\(-0.875594\pi\)
							0.924592	−	0.380959i	\(-0.124406\pi\)
\(37\)	1.27899e8	−	6.42990e8i	0.303219	−	1.52439i	−0.465643	−	0.884972i	\(-0.654177\pi\)
							0.768863	−	0.639414i	\(-0.220823\pi\)
\(41\)	6.56560e8	−	2.71956e8i	0.885040	−	0.366596i	0.106591	−	0.994303i	\(-0.466007\pi\)
							0.778449	+	0.627707i	\(0.216007\pi\)
\(43\)	−8.77342e7	+	5.86221e7i	−0.0910107	+	0.0608114i	−0.600243	−	0.799818i	\(-0.704929\pi\)
							0.509232	+	0.860629i	\(0.329929\pi\)
\(47\)	−1.63246e9	+	1.63246e9i	−1.03826	+	1.03826i	−0.0390181	+	0.999239i	\(0.512423\pi\)
							−0.999239	+	0.0390181i	\(0.987577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.i.a.5.4	✓	696
64.13	even	16	inner	64.12.i.a.13.4	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.12.i.a.5.4	✓	696	1.1	even	1	trivial
64.12.i.a.13.4	yes	696	64.13	even	16	inner