Embedded newform 32.9.h.a.19.20

• Label: 32.9.h.a.19.20
• 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			19.20
Character	\(\chi\)	\(=\)	32.19
Dual form			32.9.h.a.27.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.84160 - 15.2499i) q^{2} +(-88.5555 - 36.6809i) q^{3} +(-209.118 - 147.668i) q^{4} +(-260.598 + 107.943i) q^{5} +(-988.129 + 1172.87i) q^{6} +(-322.108 - 322.108i) q^{7} +(-3264.38 + 2474.08i) q^{8} +(1857.25 + 1857.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.84160	−	15.2499i	0.302600	−	0.953118i
							\(3\)	−88.5555	−	36.6809i	−1.09328	−	0.452850i	−0.238129	−	0.971234i	\(-0.576534\pi\)
−0.855148	+	0.518383i	\(0.826534\pi\)
\(5\)	−260.598	+	107.943i	−0.416957	+	0.172709i	−0.581292	−	0.813695i	\(-0.697452\pi\)
							0.164334	+	0.986405i	\(0.447452\pi\)
\(7\)	−322.108	−	322.108i	−0.134156	−	0.134156i	0.636840	−	0.770996i	\(-0.280241\pi\)
							−0.770996	+	0.636840i	\(0.780241\pi\)
\(11\)	7282.70	−	3016.59i	0.497418	−	0.206037i	−0.119847	−	0.992792i	\(-0.538240\pi\)
							0.617266	+	0.786755i	\(0.288240\pi\)
\(13\)	14936.4	+	6186.84i	0.522963	+	0.216618i	0.628518	−	0.777795i	\(-0.283662\pi\)
							−0.105555	+	0.994413i	\(0.533662\pi\)
\(17\)			8845.88i			0.105912i	0.998597	+	0.0529560i	\(0.0168643\pi\)
							−0.998597	+	0.0529560i	\(0.983136\pi\)
\(19\)	8396.90	−	20271.9i	0.0644325	−	0.155554i	−0.888384	−	0.459102i	\(-0.848171\pi\)
							0.952816	+	0.303548i	\(0.0981713\pi\)
\(23\)	112450.	−	112450.i	0.401836	−	0.401836i	−0.477044	−	0.878880i	\(-0.658292\pi\)
							0.878880	+	0.477044i	\(0.158292\pi\)
\(29\)	−440553.	+	1.06359e6i	−0.622883	+	1.50377i	0.225421	+	0.974261i	\(0.427624\pi\)
							−0.848304	+	0.529510i	\(0.822376\pi\)
\(31\)			214501.i			0.232264i	0.993234	+	0.116132i	\(0.0370496\pi\)
							−0.993234	+	0.116132i	\(0.962950\pi\)
\(37\)	−2.60209e6	+	1.07782e6i	−1.38840	+	0.575096i	−0.946715	−	0.322072i	\(-0.895621\pi\)
							−0.441689	+	0.897168i	\(0.645621\pi\)
\(41\)	−1.00999e6	−	1.00999e6i	−0.357424	−	0.357424i	0.505439	−	0.862863i	\(-0.331331\pi\)
							−0.862863	+	0.505439i	\(0.831331\pi\)
\(43\)	308337.	−	127717.i	0.0901887	−	0.0373574i	−0.337133	−	0.941457i	\(-0.609457\pi\)
							0.427322	+	0.904100i	\(0.359457\pi\)
\(47\)	−4.09007e6			−0.838184			−0.419092	−	0.907944i	\(-0.637651\pi\)
							−0.419092	+	0.907944i	\(0.637651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.19.20	✓	124
32.27	odd	8	inner	32.9.h.a.27.20	yes	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.19.20	✓	124	1.1	even	1	trivial
32.9.h.a.27.20	yes	124	32.27	odd	8	inner