Embedded newform 32.9.h.a.19.16

• Label: 32.9.h.a.19.16
• Level: $32$
• Weight: $9$
• Character: 32.19
• 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			19.16
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00882713 + 16.0000i) q^{2} +(-5.58723 - 2.31431i) q^{3} +(-256.000 + 0.282468i) q^{4} +(-929.188 + 384.882i) q^{5} +(36.9796 - 89.4161i) q^{6} +(2407.71 + 2407.71i) q^{7} +(-6.77923 - 4095.99i) q^{8} +(-4613.47 - 4613.47i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00882713	+	16.0000i	0.000551695	+	1.00000i
							\(3\)	−5.58723	−	2.31431i	−0.0689781	−	0.0285717i	0.347928	−	0.937521i	\(-0.386885\pi\)
−0.416906	+	0.908950i	\(0.636885\pi\)
\(5\)	−929.188	+	384.882i	−1.48670	+	0.615812i	−0.970596	−	0.240713i	\(-0.922619\pi\)
							−0.516105	+	0.856525i	\(0.672619\pi\)
\(7\)	2407.71	+	2407.71i	1.00280	+	1.00280i	0.999996	+	0.00279994i	\(0.000891251\pi\)
							0.00279994	+	0.999996i	\(0.499109\pi\)
\(11\)	14869.5	−	6159.13i	1.01560	−	0.420677i	0.188108	−	0.982148i	\(-0.439765\pi\)
							0.827496	+	0.561472i	\(0.189765\pi\)
\(13\)	−17360.0	−	7190.76i	−0.607823	−	0.251768i	0.0574747	−	0.998347i	\(-0.481695\pi\)
							−0.665297	+	0.746579i	\(0.731695\pi\)
\(17\)		−	103083.i		−	1.23421i	−0.786880	−	0.617106i	\(-0.788305\pi\)
							0.786880	−	0.617106i	\(-0.211695\pi\)
\(19\)	10423.6	−	25164.8i	0.0799841	−	0.193099i	−0.878829	−	0.477137i	\(-0.841674\pi\)
							0.958813	+	0.284039i	\(0.0916744\pi\)
\(23\)	−76384.4	+	76384.4i	−0.272956	+	0.272956i	−0.830289	−	0.557333i	\(-0.811825\pi\)
							0.557333	+	0.830289i	\(0.311825\pi\)
\(29\)	−334415.	+	807349.i	−0.472818	+	1.14148i	0.490095	+	0.871669i	\(0.336962\pi\)
							−0.962913	+	0.269813i	\(0.913038\pi\)
\(31\)		−	1.25154e6i		−	1.35518i	−0.735441	−	0.677589i	\(-0.763025\pi\)
							0.735441	−	0.677589i	\(-0.236975\pi\)
\(37\)	−1.89485e6	+	784875.i	−1.01104	+	0.418787i	−0.825835	−	0.563912i	\(-0.809296\pi\)
							−0.185207	+	0.982700i	\(0.559296\pi\)
\(41\)	−2.94229e6	−	2.94229e6i	−1.04124	−	1.04124i	−0.999112	−	0.0421266i	\(-0.986587\pi\)
							−0.0421266	−	0.999112i	\(-0.513413\pi\)
\(43\)	−2.04274e6	+	846130.i	−0.597501	+	0.247493i	−0.660874	−	0.750497i	\(-0.729814\pi\)
							0.0633729	+	0.997990i	\(0.479814\pi\)
\(47\)	2.68537e6			0.550317			0.275158	−	0.961399i	\(-0.411270\pi\)
							0.275158	+	0.961399i	\(0.411270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.16	✓	124
32.27	odd	8	inner	32.9.h.a.27.16	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.16	✓	124	1.1	even	1	trivial
32.9.h.a.27.16	yes	124	32.27	odd	8	inner