Embedded newform 430.2.t.a.9.2

• Label: 430.2.t.a.9.2
• 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			9.2
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(-0.362457 + 2.40475i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-2.23136 + 0.145098i) q^{5} +(-1.21595 - 2.10610i) q^{6} +(-1.90024 - 1.09710i) q^{7} +(0.433884 + 0.900969i) q^{8} +(-2.78472 - 0.858971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.362457	+	2.40475i	−0.209265	+	1.38838i	0.599559	+	0.800331i	\(0.295343\pi\)
−0.808824	+	0.588051i	\(0.799895\pi\)
\(5\)	−2.23136	+	0.145098i	−0.997892	+	0.0648896i
							\(7\)	−1.90024	−	1.09710i	−0.718222	−	0.414666i	0.0958759	−	0.995393i	\(-0.469435\pi\)
−0.814098	+	0.580728i	\(0.802768\pi\)
\(11\)	−0.144527	−	0.633213i	−0.0435764	−	0.190921i	0.948455	−	0.316912i	\(-0.102646\pi\)
							−0.992031	+	0.125991i	\(0.959789\pi\)
\(13\)	1.08194	+	0.0810799i	0.300075	+	0.0224875i	0.223917	−	0.974608i	\(-0.428116\pi\)
							0.0761581	+	0.997096i	\(0.475735\pi\)
\(17\)	−3.80514	−	5.58112i	−0.922883	−	1.35362i	−0.935360	−	0.353698i	\(-0.884924\pi\)
							0.0124766	−	0.999922i	\(-0.496028\pi\)
\(19\)	0.0371771	−	0.0114676i	0.00852902	−	0.00263085i	−0.290488	−	0.956879i	\(-0.593818\pi\)
							0.299017	+	0.954248i	\(0.403341\pi\)
\(23\)	−0.963167	−	1.03805i	−0.200834	−	0.216448i	0.624602	−	0.780943i	\(-0.285261\pi\)
							−0.825436	+	0.564495i	\(0.809071\pi\)
\(29\)	−0.249536	+	0.0376115i	−0.0463376	+	0.00698427i	−0.172170	−	0.985067i	\(-0.555078\pi\)
							0.125833	+	0.992051i	\(0.459840\pi\)
\(31\)	0.101051	−	0.257474i	0.0181493	−	0.0462438i	−0.921512	−	0.388350i	\(-0.873045\pi\)
							0.939661	+	0.342106i	\(0.111140\pi\)
\(37\)	3.45407	−	1.99421i	0.567845	−	0.327846i	−0.188443	−	0.982084i	\(-0.560344\pi\)
							0.756288	+	0.654239i	\(0.227011\pi\)
\(41\)	−1.45660	−	1.82651i	−0.227482	−	0.285254i	0.654971	−	0.755654i	\(-0.272681\pi\)
							−0.882453	+	0.470401i	\(0.844109\pi\)
\(43\)	6.51416	+	0.752180i	0.993399	+	0.114706i
							\(47\)	−9.92138	−	2.26449i	−1.44718	−	0.330310i	−0.574462	−	0.818531i	\(-0.694789\pi\)
−0.872720	+	0.488221i	\(0.837646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.2	✓	264
5.4	even	2	inner	430.2.t.a.9.21	yes	264
43.24	even	21	inner	430.2.t.a.239.21	yes	264
215.24	even	42	inner	430.2.t.a.239.2	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.2	✓	264	1.1	even	1	trivial
430.2.t.a.9.21	yes	264	5.4	even	2	inner
430.2.t.a.239.2	yes	264	215.24	even	42	inner
430.2.t.a.239.21	yes	264	43.24	even	21	inner