Embedded newform 430.2.t.a.109.10

• Label: 430.2.t.a.109.10
• 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			109.10
Character	\(\chi\)	\(=\)	430.109
Dual form			430.2.t.a.359.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(1.96806 - 0.772405i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-1.16870 - 1.90634i) q^{5} +(-1.05710 + 1.83095i) q^{6} +(2.54761 - 1.47086i) q^{7} +(0.433884 + 0.900969i) q^{8} +(1.07748 - 0.999758i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	1.96806	−	0.772405i	1.13626	−	0.445949i	0.278722	−	0.960372i	\(-0.410089\pi\)
0.857536	+	0.514423i	\(0.171994\pi\)
\(5\)	−1.16870	−	1.90634i	−0.522658	−	0.852542i
							\(7\)	2.54761	−	1.47086i	0.962907	−	0.555935i	0.0658405	−	0.997830i	\(-0.479027\pi\)
0.897067	+	0.441896i	\(0.145694\pi\)
\(11\)	−0.994852	−	4.35873i	−0.299959	−	1.31421i	−0.870187	−	0.492721i	\(-0.836002\pi\)
							0.570228	−	0.821486i	\(-0.306855\pi\)
\(13\)	−2.72065	+	3.99046i	−0.754572	+	1.10675i	0.236264	+	0.971689i	\(0.424077\pi\)
							−0.990837	+	0.135066i	\(0.956875\pi\)
\(17\)	3.91864	−	0.293662i	0.950411	−	0.0712234i	0.409506	−	0.912307i	\(-0.365701\pi\)
							0.540905	+	0.841084i	\(0.318082\pi\)
\(19\)	−2.66622	−	2.47389i	−0.611673	−	0.567549i	0.312312	−	0.949980i	\(-0.398897\pi\)
							−0.923984	+	0.382430i	\(0.875087\pi\)
\(23\)	1.37332	−	4.45220i	0.286357	−	0.928347i	−0.692083	−	0.721818i	\(-0.743307\pi\)
							0.978440	−	0.206530i	\(-0.0662170\pi\)
\(29\)	2.29469	−	5.84677i	0.426113	−	1.08572i	−0.543138	−	0.839644i	\(-0.682764\pi\)
							0.969251	−	0.246075i	\(-0.0791409\pi\)
\(31\)	−0.792864	+	0.119505i	−0.142403	+	0.0214637i	−0.219857	−	0.975532i	\(-0.570559\pi\)
							0.0774541	+	0.996996i	\(0.475321\pi\)
\(37\)	8.92095	+	5.15051i	1.46659	+	0.846739i	0.999302	−	0.0373637i	\(-0.0118960\pi\)
							0.467293	+	0.884103i	\(0.345229\pi\)
\(41\)	3.64909	+	4.57582i	0.569892	+	0.714622i	0.980352	−	0.197256i	\(-0.0632031\pi\)
							−0.410460	+	0.911879i	\(0.634632\pi\)
\(43\)	−6.31617	+	1.76237i	−0.963207	+	0.268759i
							\(47\)	4.98439	+	1.13765i	0.727048	+	0.165944i	0.569997	−	0.821647i	\(-0.306944\pi\)
0.157051	+	0.987591i	\(0.449801\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.109.10	✓	264
5.4	even	2	inner	430.2.t.a.109.13	yes	264
43.15	even	21	inner	430.2.t.a.359.13	yes	264
215.144	even	42	inner	430.2.t.a.359.10	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.109.10	✓	264	1.1	even	1	trivial
430.2.t.a.109.13	yes	264	5.4	even	2	inner
430.2.t.a.359.10	yes	264	215.144	even	42	inner
430.2.t.a.359.13	yes	264	43.15	even	21	inner