Embedded newform 32.9.h.a.11.2

• Label: 32.9.h.a.11.2
• Level: $32$
• Weight: $9$
• Character: 32.11
• Analytic conductor: $13.036$
• Analytic rank: $0$
• Dimension: $124$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0361155220\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(31\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			11.2
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.7351 - 2.89922i) q^{2} +(-45.7664 - 110.490i) q^{3} +(239.189 + 91.2392i) q^{4} +(-3.17531 + 7.66587i) q^{5} +(399.806 + 1871.26i) q^{6} +(-1922.55 + 1922.55i) q^{7} +(-3499.15 - 2129.12i) q^{8} +(-5474.11 + 5474.11i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.7351	−	2.89922i	−0.983446	−	0.181201i
							\(3\)	−45.7664	−	110.490i	−0.565017	−	1.36407i	−0.905710	−	0.423897i	\(-0.860662\pi\)
0.340693	−	0.940174i	\(-0.389338\pi\)
\(5\)	−3.17531	+	7.66587i	−0.00508049	+	0.0122654i	−0.926399	−	0.376542i	\(-0.877113\pi\)
							0.921319	+	0.388808i	\(0.127113\pi\)
\(7\)	−1922.55	+	1922.55i	−0.800727	+	0.800727i	−0.983209	−	0.182482i	\(-0.941587\pi\)
							0.182482	+	0.983209i	\(0.441587\pi\)
\(11\)	−1037.61	+	2505.02i	−0.0708703	+	0.171096i	−0.955345	−	0.295491i	\(-0.904517\pi\)
							0.884475	+	0.466587i	\(0.154517\pi\)
\(13\)	5550.16	+	13399.3i	0.194327	+	0.469146i	0.990768	−	0.135570i	\(-0.0432866\pi\)
							−0.796441	+	0.604716i	\(0.793287\pi\)
\(17\)			27189.5i			0.325540i	0.986664	+	0.162770i	\(0.0520429\pi\)
							−0.986664	+	0.162770i	\(0.947957\pi\)
\(19\)	111157.	−	46042.5i	0.852944	−	0.353301i	0.0870002	−	0.996208i	\(-0.472272\pi\)
							0.765944	+	0.642907i	\(0.222272\pi\)
\(23\)	−111649.	−	111649.i	−0.398974	−	0.398974i	0.478897	−	0.877871i	\(-0.341037\pi\)
							−0.877871	+	0.478897i	\(0.841037\pi\)
\(29\)	1.15183e6	−	477105.i	1.62854	−	0.674562i	0.633470	−	0.773767i	\(-0.281630\pi\)
							0.995067	+	0.0992054i	\(0.0316301\pi\)
\(31\)		−	1.18054e6i		−	1.27830i	−0.769082	−	0.639150i	\(-0.779286\pi\)
							0.769082	−	0.639150i	\(-0.220714\pi\)
\(37\)	−1.00485e6	+	2.42592e6i	−0.536160	+	1.29440i	0.391225	+	0.920295i	\(0.372051\pi\)
							−0.927385	+	0.374109i	\(0.877949\pi\)
\(41\)	−2.12875e6	+	2.12875e6i	−0.753336	+	0.753336i	−0.975100	−	0.221764i	\(-0.928819\pi\)
							0.221764	+	0.975100i	\(0.428819\pi\)
\(43\)	−1.75182e6	+	4.22928e6i	−0.512409	+	1.23706i	0.430069	+	0.902796i	\(0.358489\pi\)
							−0.942478	+	0.334268i	\(0.891511\pi\)
\(47\)	5.90520e6			1.21016			0.605080	−	0.796164i	\(-0.293141\pi\)
							0.605080	+	0.796164i	\(0.293141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.2	yes	124
32.3	odd	8	inner	32.9.h.a.3.2	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.2	✓	124	32.3	odd	8	inner
32.9.h.a.11.2	yes	124	1.1	even	1	trivial