Embedded newform 430.2.t.a.109.14

• Label: 430.2.t.a.109.14
• Level: $430$
• Weight: $2$
• Character: 430.109
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• 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			109.14
Character	\(\chi\)	\(=\)	430.109
Dual form			430.2.t.a.359.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.781831 - 0.623490i) q^{2} +(-1.57478 + 0.618055i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-1.72517 - 1.42260i) q^{5} +(-0.845861 + 1.46507i) q^{6} +(-0.731645 + 0.422415i) q^{7} +(-0.433884 - 0.900969i) q^{8} +(-0.101218 + 0.0939162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{8}{21}\right)\)

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
							\(3\)	−1.57478	+	0.618055i	−0.909199	+	0.356834i	−0.773412	−	0.633904i	\(-0.781452\pi\)
−0.135787	+	0.990738i	\(0.543356\pi\)
\(5\)	−1.72517	−	1.42260i	−0.771519	−	0.636207i
							\(7\)	−0.731645	+	0.422415i	−0.276536	+	0.159658i	−0.631854	−	0.775087i	\(-0.717706\pi\)
0.355318	+	0.934745i	\(0.384372\pi\)
\(11\)	1.27846	+	5.60130i	0.385470	+	1.68885i	0.680000	+	0.733212i	\(0.261980\pi\)
							−0.294530	+	0.955642i	\(0.595163\pi\)
\(13\)	−1.25534	+	1.84125i	−0.348170	+	0.510671i	−0.959880	−	0.280411i	\(-0.909529\pi\)
							0.611710	+	0.791082i	\(0.290482\pi\)
\(17\)	−2.04920	+	0.153567i	−0.497005	+	0.0372454i	−0.320875	−	0.947122i	\(-0.603977\pi\)
							−0.176130	+	0.984367i	\(0.556358\pi\)
\(19\)	−1.51961	−	1.40999i	−0.348623	−	0.323475i	0.486352	−	0.873763i	\(-0.338327\pi\)
							−0.834975	+	0.550288i	\(0.814518\pi\)
\(23\)	−2.15041	+	6.97146i	−0.448391	+	1.45365i	0.396164	+	0.918180i	\(0.370341\pi\)
							−0.844555	+	0.535469i	\(0.820135\pi\)
\(29\)	0.229736	−	0.585357i	0.0426609	−	0.108698i	−0.907955	−	0.419067i	\(-0.862357\pi\)
							0.950616	+	0.310369i	\(0.100453\pi\)
\(31\)	−4.82154	+	0.726729i	−0.865973	+	0.130524i	−0.566981	−	0.823731i	\(-0.691889\pi\)
							−0.298992	+	0.954255i	\(0.596651\pi\)
\(37\)	−0.338150	−	0.195231i	−0.0555915	−	0.0320958i	0.471947	−	0.881627i	\(-0.343551\pi\)
							−0.527538	+	0.849531i	\(0.676885\pi\)
\(41\)	−5.05655	−	6.34071i	−0.789700	−	0.990253i	−0.999921	−	0.0125990i	\(-0.995990\pi\)
							0.210220	−	0.977654i	\(-0.432582\pi\)
\(43\)	−6.35603	−	1.61272i	−0.969286	−	0.245938i
							\(47\)	6.26877	+	1.43081i	0.914394	+	0.208704i	0.653732	−	0.756726i	\(-0.273202\pi\)
0.260662	+	0.965430i	\(0.416059\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.109.14	yes	264
5.4	even	2	inner	430.2.t.a.109.9	✓	264
43.15	even	21	inner	430.2.t.a.359.9	yes	264
215.144	even	42	inner	430.2.t.a.359.14	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.109.9	✓	264	5.4	even	2	inner
430.2.t.a.109.14	yes	264	1.1	even	1	trivial
430.2.t.a.359.9	yes	264	43.15	even	21	inner
430.2.t.a.359.14	yes	264	215.144	even	42	inner