Embedded newform 32.9.h.a.11.5

• Label: 32.9.h.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.4978 + 8.59129i) q^{2} +(-45.3301 - 109.437i) q^{3} +(108.379 - 231.927i) q^{4} +(105.398 - 254.454i) q^{5} +(1552.06 + 1087.71i) q^{6} +(3075.72 - 3075.72i) q^{7} +(529.671 + 4061.61i) q^{8} +(-5282.22 + 5282.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.4978	+	8.59129i	−0.843610	+	0.536956i
							\(3\)	−45.3301	−	109.437i	−0.559631	−	1.35107i	−0.910059	−	0.414479i	\(-0.863964\pi\)
0.350427	−	0.936590i	\(-0.386036\pi\)
\(5\)	105.398	−	254.454i	0.168638	−	0.407127i	−0.816856	−	0.576842i	\(-0.804285\pi\)
							0.985493	+	0.169715i	\(0.0542847\pi\)
\(7\)	3075.72	−	3075.72i	1.28101	−	1.28101i	0.340923	−	0.940091i	\(-0.389260\pi\)
							0.940091	−	0.340923i	\(-0.110740\pi\)
\(11\)	6442.72	−	15554.1i	0.440046	−	1.06237i	−0.535886	−	0.844291i	\(-0.680022\pi\)
							0.975932	−	0.218075i	\(-0.0699778\pi\)
\(13\)	9249.65	+	22330.6i	0.323856	+	0.781857i	0.999023	+	0.0441924i	\(0.0140715\pi\)
							−0.675167	+	0.737665i	\(0.735929\pi\)
\(17\)		−	152834.i		−	1.82989i	−0.403577	−	0.914946i	\(-0.632233\pi\)
							0.403577	−	0.914946i	\(-0.367767\pi\)
\(19\)	54895.1	−	22738.3i	0.421230	−	0.174479i	−0.161992	−	0.986792i	\(-0.551792\pi\)
							0.583222	+	0.812313i	\(0.301792\pi\)
\(23\)	−70961.1	−	70961.1i	−0.253577	−	0.253577i	0.568859	−	0.822435i	\(-0.307385\pi\)
							−0.822435	+	0.568859i	\(0.807385\pi\)
\(29\)	−889486.	+	368437.i	−1.25761	+	0.520921i	−0.909176	−	0.416412i	\(-0.863288\pi\)
							−0.348437	+	0.937332i	\(0.613288\pi\)
\(31\)			1.25711e6i			1.36122i	0.732647	+	0.680608i	\(0.238284\pi\)
							−0.732647	+	0.680608i	\(0.761716\pi\)
\(37\)	−272612.	+	658144.i	−0.145458	+	0.351167i	−0.979770	−	0.200125i	\(-0.935865\pi\)
							0.834312	+	0.551292i	\(0.185865\pi\)
\(41\)	668372.	−	668372.i	0.236528	−	0.236528i	−0.578883	−	0.815411i	\(-0.696511\pi\)
							0.815411	+	0.578883i	\(0.196511\pi\)
\(43\)	−895890.	+	2.16287e6i	−0.262048	+	0.632640i	−0.999065	−	0.0432343i	\(-0.986234\pi\)
							0.737017	+	0.675874i	\(0.236234\pi\)
\(47\)	4.42592e6			0.907009			0.453505	−	0.891254i	\(-0.350174\pi\)
							0.453505	+	0.891254i	\(0.350174\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.5	yes	124
32.3	odd	8	inner	32.9.h.a.3.5	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.5	✓	124	32.3	odd	8	inner
32.9.h.a.11.5	yes	124	1.1	even	1	trivial