Embedded newform 430.2.r.a.323.14

• Label: 430.2.r.a.323.14
• Level: $430$
• Weight: $2$
• Character: 430.323
• 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.r (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			323.14
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(-0.655098 + 1.87216i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-1.30201 + 1.81791i) q^{5} -1.98347 q^{6} +(2.99846 + 2.99846i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(-0.730349 - 0.582434i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.330279	+	0.943883i	0.233543	+	0.667426i
							\(3\)	−0.655098	+	1.87216i	−0.378221	+	1.08089i	0.584732	+	0.811226i	\(0.301200\pi\)
−0.962953	+	0.269668i	\(0.913086\pi\)
\(5\)	−1.30201	+	1.81791i	−0.582274	+	0.812992i
							\(7\)	2.99846	+	2.99846i	1.13331	+	1.13331i	0.989623	+	0.143688i	\(0.0458960\pi\)
0.143688	+	0.989623i	\(0.454104\pi\)
\(11\)	−2.11398	+	2.65085i	−0.637390	+	0.799262i	−0.990674	−	0.136254i	\(-0.956494\pi\)
							0.353284	+	0.935516i	\(0.385065\pi\)
\(13\)	3.59605	−	5.72308i	0.997365	−	1.58730i	0.203618	−	0.979050i	\(-0.434730\pi\)
							0.793747	−	0.608248i	\(-0.208127\pi\)
\(17\)	1.56758	+	2.49479i	0.380194	+	0.605076i	0.980928	−	0.194374i	\(-0.0622675\pi\)
							−0.600733	+	0.799450i	\(0.705125\pi\)
\(19\)	−4.21640	−	5.28720i	−0.967309	−	1.21297i	−0.977049	−	0.213014i	\(-0.931672\pi\)
							0.00974048	−	0.999953i	\(-0.496899\pi\)
\(23\)	−0.239409	+	0.0269749i	−0.0499203	+	0.00562467i	−0.136889	−	0.990586i	\(-0.543710\pi\)
							0.0869690	+	0.996211i	\(0.472282\pi\)
\(29\)	2.78148	+	1.33949i	0.516508	+	0.248737i	0.673931	−	0.738794i	\(-0.264604\pi\)
							−0.157423	+	0.987531i	\(0.550319\pi\)
\(31\)	6.96224	+	3.35284i	1.25045	+	0.602187i	0.937634	−	0.347625i	\(-0.113012\pi\)
							0.312821	+	0.949812i	\(0.398726\pi\)
\(37\)	−3.20464	−	3.20464i	−0.526840	−	0.526840i	0.392789	−	0.919629i	\(-0.371510\pi\)
							−0.919629	+	0.392789i	\(0.871510\pi\)
\(41\)	1.86251	+	0.896935i	0.290874	+	0.140078i	0.573631	−	0.819114i	\(-0.305534\pi\)
							−0.282756	+	0.959192i	\(0.591249\pi\)
\(43\)	−6.55734	+	0.0357397i	−0.999985	+	0.00545026i
							\(47\)	1.16391	+	0.131141i	0.169773	+	0.0191289i	0.196442	−	0.980515i	\(-0.437061\pi\)
−0.0266685	+	0.999644i	\(0.508490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.14	yes	264
5.2	odd	4	inner	430.2.r.a.237.9	yes	264
43.2	odd	14	inner	430.2.r.a.303.9	yes	264
215.2	even	28	inner	430.2.r.a.217.14	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.14	✓	264	215.2	even	28	inner
430.2.r.a.237.9	yes	264	5.2	odd	4	inner
430.2.r.a.303.9	yes	264	43.2	odd	14	inner
430.2.r.a.323.14	yes	264	1.1	even	1	trivial