Embedded newform 89.9.d.a.12.4

• Label: 89.9.d.a.12.4
• Level: $89$
• Weight: $9$
• Character: 89.12
• Analytic conductor: $36.257$
• Analytic rank: $0$
• Dimension: $236$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	89.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(36.2566962954\)
Analytic rank:	\(0\)
Dimension:	\(236\)
Relative dimension:	\(59\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			12.4
Character	\(\chi\)	\(=\)	89.12
Dual form			89.9.d.a.52.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-28.5712 q^{2} +(-56.6127 - 136.675i) q^{3} +560.314 q^{4} +(-23.2205 - 23.2205i) q^{5} +(1617.49 + 3904.98i) q^{6} +(166.667 + 402.371i) q^{7} -8694.63 q^{8} +(-10835.8 + 10835.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 236 q - 8 q^{2} + 220 q^{3} + 29688 q^{4} + 332 q^{5} - 2052 q^{6} - 4 q^{7} - 2056 q^{8} + 14504 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/89\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−28.5712			−1.78570			−0.892851	−	0.450353i	\(-0.851298\pi\)
							−0.892851	+	0.450353i	\(0.851298\pi\)
\(3\)	−56.6127	−	136.675i	−0.698922	−	1.68735i	−0.725982	−	0.687714i	\(-0.758614\pi\)
							0.0270593	−	0.999634i	\(-0.491386\pi\)
\(5\)	−23.2205	−	23.2205i	−0.0371529	−	0.0371529i	0.688286	−	0.725439i	\(-0.258363\pi\)
							−0.725439	+	0.688286i	\(0.758363\pi\)
\(7\)	166.667	+	402.371i	0.0694158	+	0.167585i	0.954780	−	0.297314i	\(-0.0960908\pi\)
							−0.885364	+	0.464899i	\(0.846091\pi\)
\(11\)		−	4054.68i		−	0.276940i	−0.990367	−	0.138470i	\(-0.955782\pi\)
							0.990367	−	0.138470i	\(-0.0442184\pi\)
\(13\)	−5797.96	+	13997.5i	−0.203003	+	0.490092i	−0.992291	−	0.123931i	\(-0.960450\pi\)
							0.789288	+	0.614023i	\(0.210450\pi\)
\(17\)	12849.2	+	12849.2i	0.153844	+	0.153844i	0.779833	−	0.625988i	\(-0.215304\pi\)
							−0.625988	+	0.779833i	\(0.715304\pi\)
\(19\)	44242.3	+	18325.8i	0.339487	+	0.140620i	0.545912	−	0.837842i	\(-0.316183\pi\)
							−0.206426	+	0.978462i	\(0.566183\pi\)
\(23\)	156123.	+	376913.i	0.557898	+	1.34688i	0.911427	+	0.411461i	\(0.134981\pi\)
							−0.353530	+	0.935423i	\(0.615019\pi\)
\(29\)	−164760.	−	68245.6i	−0.232948	−	0.0964901i	0.263156	−	0.964753i	\(-0.415237\pi\)
							−0.496104	+	0.868263i	\(0.665237\pi\)
\(31\)	277448.	−	669818.i	0.300424	−	0.725287i	−0.699519	−	0.714614i	\(-0.746603\pi\)
							0.999943	−	0.0106733i	\(-0.00339748\pi\)
\(37\)	1.90088e6	+	787371.i	1.01426	+	0.420119i	0.827006	−	0.562193i	\(-0.190042\pi\)
							0.187251	+	0.982312i	\(0.440042\pi\)
\(41\)	−842353.	+	348914.i	−0.298098	+	0.123476i	−0.526719	−	0.850039i	\(-0.676578\pi\)
							0.228622	+	0.973515i	\(0.426578\pi\)
\(43\)	5.99283e6	−	2.48231e6i	1.75290	−	0.726077i	0.755416	−	0.655246i	\(-0.227435\pi\)
							0.997488	−	0.0708305i	\(-0.0225649\pi\)
\(47\)	−2.22932e6	−	2.22932e6i	−0.456857	−	0.456857i	0.440765	−	0.897622i	\(-0.354707\pi\)
							−0.897622	+	0.440765i	\(0.854707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.9.d.a.12.4	✓	236
89.52	odd	8	inner	89.9.d.a.52.4	yes	236

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.9.d.a.12.4	✓	236	1.1	even	1	trivial
89.9.d.a.52.4	yes	236	89.52	odd	8	inner