Embedded newform 430.2.t.a.9.7

• Label: 430.2.t.a.9.7
• 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			9.7
Character	\(\chi\)	\(=\)	430.9
Dual form			430.2.t.a.239.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.781831 + 0.623490i) q^{2} +(0.0684663 - 0.454244i) q^{3} +(0.222521 - 0.974928i) q^{4} +(-0.0829298 - 2.23453i) q^{5} +(0.229688 + 0.397830i) q^{6} +(-1.76000 - 1.01613i) q^{7} +(0.433884 + 0.900969i) q^{8} +(2.66507 + 0.822065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q + 44 q^{4} + 4 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \)

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
							\(3\)	0.0684663	−	0.454244i	0.0395290	−	0.262258i	−0.960333	−	0.278855i	\(-0.910045\pi\)
0.999862	+	0.0165970i	\(0.00528322\pi\)
\(5\)	−0.0829298	−	2.23453i	−0.0370873	−	0.999312i
							\(7\)	−1.76000	−	1.01613i	−0.665216	−	0.384062i	0.129046	−	0.991639i	\(-0.458809\pi\)
−0.794261	+	0.607576i	\(0.792142\pi\)
\(11\)	−0.283937	−	1.24401i	−0.0856101	−	0.375082i	0.913915	−	0.405906i	\(-0.133044\pi\)
							−0.999525	+	0.0308238i	\(0.990187\pi\)
\(13\)	−6.39647	−	0.479349i	−1.77406	−	0.132947i	−0.852886	−	0.522097i	\(-0.825150\pi\)
							−0.921175	+	0.389150i	\(0.872769\pi\)
\(17\)	−0.326652	−	0.479110i	−0.0792247	−	0.116201i	0.784581	−	0.620026i	\(-0.212878\pi\)
							−0.863806	+	0.503825i	\(0.831926\pi\)
\(19\)	1.37371	−	0.423734i	0.315151	−	0.0972111i	−0.133145	−	0.991097i	\(-0.542507\pi\)
							0.448295	+	0.893885i	\(0.352031\pi\)
\(23\)	−2.88150	−	3.10552i	−0.600834	−	0.647545i	0.357258	−	0.934006i	\(-0.383712\pi\)
							−0.958092	+	0.286461i	\(0.907521\pi\)
\(29\)	−10.3159	+	1.55488i	−1.91562	+	0.288733i	−0.994849	−	0.101367i	\(-0.967678\pi\)
							−0.920772	+	0.390101i	\(0.872440\pi\)
\(31\)	3.00552	−	7.65794i	0.539807	−	1.37541i	−0.357464	−	0.933927i	\(-0.616359\pi\)
							0.897271	−	0.441480i	\(-0.145546\pi\)
\(37\)	2.78144	−	1.60586i	0.457265	−	0.264002i	−0.253628	−	0.967302i	\(-0.581624\pi\)
							0.710894	+	0.703299i	\(0.248291\pi\)
\(41\)	4.21199	+	5.28166i	0.657802	+	0.824857i	0.993102	−	0.117254i	\(-0.0374092\pi\)
							−0.335300	+	0.942111i	\(0.608838\pi\)
\(43\)	5.21941	−	3.96961i	0.795952	−	0.605360i
							\(47\)	7.97653	+	1.82059i	1.16350	+	0.265560i	0.760312	−	0.649558i	\(-0.225046\pi\)
0.403184	+	0.915119i	\(0.367903\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.t.a.9.7	✓	264
5.4	even	2	inner	430.2.t.a.9.16	yes	264
43.24	even	21	inner	430.2.t.a.239.16	yes	264
215.24	even	42	inner	430.2.t.a.239.7	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.t.a.9.7	✓	264	1.1	even	1	trivial
430.2.t.a.9.16	yes	264	5.4	even	2	inner
430.2.t.a.239.7	yes	264	215.24	even	42	inner
430.2.t.a.239.16	yes	264	43.24	even	21	inner