Embedded newform 430.2.x.a.3.15

• Label: 430.2.x.a.3.15
• Level: $430$
• Weight: $2$
• Character: 430.3
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 430 = 2 \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	430.x (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43356728692\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(22\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			3.15
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(-0.667495 - 0.775643i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-1.70213 + 1.45008i) q^{5} +(0.511657 - 0.886215i) q^{6} +(0.854652 - 3.18961i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(0.291054 - 1.93102i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{42}\right)\)

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
							\(3\)	−0.667495	−	0.775643i	−0.385378	−	0.447818i	0.531456	−	0.847086i	\(-0.321645\pi\)
−0.916834	+	0.399268i	\(0.869264\pi\)
\(5\)	−1.70213	+	1.45008i	−0.761217	+	0.648498i
							\(7\)	0.854652	−	3.18961i	0.323028	−	1.20556i	−0.593251	−	0.805018i	\(-0.702156\pi\)
0.916279	−	0.400540i	\(-0.131178\pi\)
\(11\)	−0.490602	+	0.615195i	−0.147922	+	0.185488i	−0.850272	−	0.526343i	\(-0.823563\pi\)
							0.702350	+	0.711832i	\(0.252134\pi\)
\(13\)	5.50614	+	0.206025i	1.52713	+	0.0571411i	0.787686	−	0.616077i	\(-0.211279\pi\)
							0.739444	+	0.673219i	\(0.235089\pi\)
\(17\)	2.45908	−	4.65281i	0.596415	−	1.12847i	−0.382452	−	0.923975i	\(-0.624920\pi\)
							0.978867	−	0.204497i	\(-0.0655560\pi\)
\(19\)	4.24673	−	0.640092i	0.974268	−	0.146847i	0.357432	−	0.933939i	\(-0.383652\pi\)
							0.616836	+	0.787092i	\(0.288414\pi\)
\(23\)	−1.89147	+	4.33530i	−0.394400	+	0.903973i	0.600167	+	0.799874i	\(0.295101\pi\)
							−0.994567	+	0.104099i	\(0.966804\pi\)
\(29\)	−0.484080	−	6.45960i	−0.0898914	−	1.19952i	−0.841630	−	0.540055i	\(-0.818403\pi\)
							0.751738	−	0.659462i	\(-0.229216\pi\)
\(31\)	−3.21891	+	2.19461i	−0.578133	+	0.394164i	−0.816786	−	0.576941i	\(-0.804246\pi\)
							0.238653	+	0.971105i	\(0.423294\pi\)
\(37\)	9.50944	−	2.54805i	1.56334	−	0.418896i	0.629623	−	0.776901i	\(-0.283209\pi\)
							0.933720	+	0.358004i	\(0.116543\pi\)
\(41\)	−7.43566	−	3.58082i	−1.16125	−	0.559231i	−0.248859	−	0.968540i	\(-0.580055\pi\)
							−0.912396	+	0.409309i	\(0.865770\pi\)
\(43\)	3.11452	−	5.77059i	0.474961	−	0.880007i
							\(47\)	−1.64180	−	0.184987i	−0.239481	−	0.0269831i	−0.00859198	−	0.999963i	\(-0.502735\pi\)
−0.230889	+	0.972980i	\(0.574164\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.15	✓	528
5.2	odd	4	inner	430.2.x.a.347.4	yes	528
43.29	odd	42	inner	430.2.x.a.373.4	yes	528
215.72	even	84	inner	430.2.x.a.287.15	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.15	✓	528	1.1	even	1	trivial
430.2.x.a.287.15	yes	528	215.72	even	84	inner
430.2.x.a.347.4	yes	528	5.2	odd	4	inner
430.2.x.a.373.4	yes	528	43.29	odd	42	inner