Embedded newform 32.9.h.a.19.31

• Label: 32.9.h.a.19.31
• 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			19.31
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.9998 - 0.0851190i) q^{2} +(-103.033 - 42.6778i) q^{3} +(255.986 - 2.72377i) q^{4} +(-547.329 + 226.711i) q^{5} +(-1652.14 - 674.066i) q^{6} +(2254.92 + 2254.92i) q^{7} +(4095.48 - 65.3689i) q^{8} +(4155.16 + 4155.16i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.9998	−	0.0851190i	0.999986	−	0.00531994i
							\(3\)	−103.033	−	42.6778i	−1.27202	−	0.526887i	−0.358441	−	0.933553i	\(-0.616691\pi\)
−0.913577	+	0.406666i	\(0.866691\pi\)
\(5\)	−547.329	+	226.711i	−0.875726	+	0.362737i	−0.774838	−	0.632160i	\(-0.782168\pi\)
							−0.100888	+	0.994898i	\(0.532168\pi\)
\(7\)	2254.92	+	2254.92i	0.939158	+	0.939158i	0.998252	−	0.0590941i	\(-0.0188212\pi\)
							−0.0590941	+	0.998252i	\(0.518821\pi\)
\(11\)	10315.1	−	4272.65i	0.704535	−	0.291828i	−0.00150586	−	0.999999i	\(-0.500479\pi\)
							0.706041	+	0.708171i	\(0.250479\pi\)
\(13\)	34777.1	+	14405.2i	1.21764	+	0.504365i	0.896660	−	0.442720i	\(-0.145986\pi\)
							0.320984	+	0.947085i	\(0.395986\pi\)
\(17\)			56216.8i			0.673085i	0.941668	+	0.336543i	\(0.109258\pi\)
							−0.941668	+	0.336543i	\(0.890742\pi\)
\(19\)	−13803.4	+	33324.4i	−0.105919	+	0.255710i	−0.967949	−	0.251146i	\(-0.919193\pi\)
							0.862031	+	0.506856i	\(0.169193\pi\)
\(23\)	−248581.	+	248581.i	−0.888295	+	0.888295i	−0.994359	−	0.106064i	\(-0.966175\pi\)
							0.106064	+	0.994359i	\(0.466175\pi\)
\(29\)	443895.	−	1.07166e6i	0.627608	−	1.51518i	−0.214978	−	0.976619i	\(-0.568968\pi\)
							0.842586	−	0.538561i	\(-0.181032\pi\)
\(31\)			1.50006e6i			1.62429i	0.583458	+	0.812143i	\(0.301699\pi\)
							−0.583458	+	0.812143i	\(0.698301\pi\)
\(37\)	2.06844e6	−	856777.i	1.10366	−	0.457152i	0.244912	−	0.969545i	\(-0.421241\pi\)
							0.858751	+	0.512393i	\(0.171241\pi\)
\(41\)	−2.20699e6	−	2.20699e6i	−0.781026	−	0.781026i	0.198978	−	0.980004i	\(-0.436238\pi\)
							−0.980004	+	0.198978i	\(0.936238\pi\)
\(43\)	−1.43484e6	+	594332.i	−0.419692	+	0.173842i	−0.582527	−	0.812811i	\(-0.697936\pi\)
							0.162835	+	0.986653i	\(0.447936\pi\)
\(47\)	3.09345e6			0.633945			0.316972	−	0.948435i	\(-0.397334\pi\)
							0.316972	+	0.948435i	\(0.397334\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.31	✓	124
32.27	odd	8	inner	32.9.h.a.27.31	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.31	✓	124	1.1	even	1	trivial
32.9.h.a.27.31	yes	124	32.27	odd	8	inner