Embedded newform 430.2.t.a.9.8

• Label: 430.2.t.a.9.8
• Level: $430$
• Weight: $2$
• Character: 430.9
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			9.8
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(0.225928 - 1.49894i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-2.22164 + 0.253613i) q^{5} +(0.757933 + 1.31278i) q^{6} +(3.55836 + 2.05442i) q^{7} +(0.433884 + 0.900969i) q^{8} +(0.670955 + 0.206962i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.781831	+	0.623490i	−0.552838	+	0.440874i
							\(3\)	0.225928	−	1.49894i	0.130440	−	0.865411i	−0.823910	−	0.566720i	\(-0.808212\pi\)
0.954350	−	0.298691i	\(-0.0965498\pi\)
\(5\)	−2.22164	+	0.253613i	−0.993547	+	0.113419i
							\(7\)	3.55836	+	2.05442i	1.34493	+	0.776497i	0.987527	−	0.157452i	\(-0.0503278\pi\)
0.357406	+	0.933949i	\(0.383661\pi\)
\(11\)	−0.995310	−	4.36074i	−0.300097	−	1.31481i	−0.869980	−	0.493087i	\(-0.835869\pi\)
							0.569883	−	0.821726i	\(-0.306989\pi\)
\(13\)	0.439979	+	0.0329719i	0.122028	+	0.00914475i	0.135604	−	0.990763i	\(-0.456702\pi\)
							−0.0135760	+	0.999908i	\(0.504322\pi\)
\(17\)	1.50915	+	2.21352i	0.366022	+	0.536856i	0.964431	−	0.264335i	\(-0.0851524\pi\)
							−0.598408	+	0.801191i	\(0.704200\pi\)
\(19\)	1.15995	−	0.357799i	0.266112	−	0.0820846i	−0.158828	−	0.987306i	\(-0.550771\pi\)
							0.424940	+	0.905222i	\(0.360295\pi\)
\(23\)	−2.04851	−	2.20776i	−0.427143	−	0.460351i	0.482406	−	0.875948i	\(-0.339763\pi\)
							−0.909549	+	0.415597i	\(0.863573\pi\)
\(29\)	8.36479	−	1.26079i	1.55330	−	0.234123i	0.684400	−	0.729107i	\(-0.260064\pi\)
							0.868902	+	0.494984i	\(0.164826\pi\)
\(31\)	2.63801	−	6.72155i	0.473801	−	1.20723i	−0.471744	−	0.881735i	\(-0.656375\pi\)
							0.945545	−	0.325490i	\(-0.105529\pi\)
\(37\)	−1.39795	+	0.807108i	−0.229822	+	0.132688i	−0.610490	−	0.792024i	\(-0.709027\pi\)
							0.380668	+	0.924712i	\(0.375694\pi\)
\(41\)	−1.10654	−	1.38756i	−0.172813	−	0.216701i	0.687881	−	0.725824i	\(-0.258541\pi\)
							−0.860694	+	0.509123i	\(0.829970\pi\)
\(43\)	5.68894	−	3.26128i	0.867556	−	0.497340i
							\(47\)	1.98210	+	0.452401i	0.289119	+	0.0659895i	0.364621	−	0.931156i	\(-0.381199\pi\)
−0.0755016	+	0.997146i	\(0.524056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.8	✓	264
5.4	even	2	inner	430.2.t.a.9.15	yes	264
43.24	even	21	inner	430.2.t.a.239.15	yes	264
215.24	even	42	inner	430.2.t.a.239.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.8	✓	264	1.1	even	1	trivial
430.2.t.a.9.15	yes	264	5.4	even	2	inner
430.2.t.a.239.8	yes	264	215.24	even	42	inner
430.2.t.a.239.15	yes	264	43.24	even	21	inner