Embedded newform 430.2.t.a.109.11

• Label: 430.2.t.a.109.11
• 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			109.11
Character	\(\chi\)	\(=\)	430.109
Dual form			430.2.t.a.359.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(3.00365 - 1.17884i) q^{3} +(0.222521 - 0.974928i) q^{4} +(1.44794 + 1.70396i) q^{5} +(-1.61335 + 2.79440i) q^{6} +(0.213198 - 0.123090i) q^{7} +(0.433884 + 0.900969i) q^{8} +(5.43306 - 5.04115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	3.00365	−	1.17884i	1.73416	−	0.680606i	0.734158	−	0.678979i	\(-0.237577\pi\)
0.999999	−	0.00162707i	\(-0.000517914\pi\)
\(5\)	1.44794	+	1.70396i	0.647539	+	0.762032i
							\(7\)	0.213198	−	0.123090i	0.0805812	−	0.0465236i	−0.459168	−	0.888349i	\(-0.651852\pi\)
0.539749	+	0.841826i	\(0.318519\pi\)
\(11\)	−0.183834	−	0.805431i	−0.0554281	−	0.242847i	0.939623	−	0.342212i	\(-0.111176\pi\)
							−0.995051	+	0.0993651i	\(0.968319\pi\)
\(13\)	−0.997223	+	1.46266i	−0.276580	+	0.405668i	−0.939245	−	0.343248i	\(-0.888473\pi\)
							0.662665	+	0.748916i	\(0.269425\pi\)
\(17\)	−2.16979	+	0.162603i	−0.526251	+	0.0394371i	−0.335207	−	0.942144i	\(-0.608806\pi\)
							−0.191044	+	0.981582i	\(0.561187\pi\)
\(19\)	−3.15878	−	2.93092i	−0.724673	−	0.672398i	0.228938	−	0.973441i	\(-0.426475\pi\)
							−0.953611	+	0.301043i	\(0.902665\pi\)
\(23\)	−2.70017	+	8.75374i	−0.563025	+	1.82528i	0.00169537	+	0.999999i	\(0.499460\pi\)
							−0.564720	+	0.825283i	\(0.691016\pi\)
\(29\)	2.04110	−	5.20063i	0.379023	−	0.965734i	−0.606331	−	0.795212i	\(-0.707360\pi\)
							0.985354	−	0.170522i	\(-0.0545452\pi\)
\(31\)	0.963951	−	0.145292i	0.173131	−	0.0260953i	−0.0619047	−	0.998082i	\(-0.519717\pi\)
							0.235035	+	0.971987i	\(0.424479\pi\)
\(37\)	−9.08267	−	5.24388i	−1.49318	−	0.862089i	−0.493212	−	0.869909i	\(-0.664177\pi\)
							−0.999969	+	0.00782017i	\(0.997511\pi\)
\(41\)	−2.01651	−	2.52862i	−0.314926	−	0.394904i	0.599025	−	0.800731i	\(-0.295555\pi\)
							−0.913950	+	0.405826i	\(0.866984\pi\)
\(43\)	1.99202	−	6.24755i	0.303780	−	0.952742i
							\(47\)	9.80251	+	2.23736i	1.42984	+	0.326352i	0.866213	−	0.499675i	\(-0.166547\pi\)
0.563630	+	0.826028i	\(0.309405\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.109.11	✓	264
5.4	even	2	inner	430.2.t.a.109.12	yes	264
43.15	even	21	inner	430.2.t.a.359.12	yes	264
215.144	even	42	inner	430.2.t.a.359.11	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.109.11	✓	264	1.1	even	1	trivial
430.2.t.a.109.12	yes	264	5.4	even	2	inner
430.2.t.a.359.11	yes	264	215.144	even	42	inner
430.2.t.a.359.12	yes	264	43.15	even	21	inner