Embedded newform 430.2.x.a.3.18

• Label: 430.2.x.a.3.18
• 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			3.18
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(0.447979 + 0.520562i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-0.398933 - 2.20019i) q^{5} +(-0.343391 + 0.594771i) q^{6} +(0.739536 - 2.75998i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(0.376828 - 2.50009i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

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
							\(3\)	0.447979	+	0.520562i	0.258641	+	0.300546i	0.872247	−	0.489066i	\(-0.162662\pi\)
−0.613606	+	0.789613i	\(0.710281\pi\)
\(5\)	−0.398933	−	2.20019i	−0.178408	−	0.983957i
							\(7\)	0.739536	−	2.75998i	0.279518	−	1.04318i	−0.673234	−	0.739429i	\(-0.735096\pi\)
0.952753	−	0.303747i	\(-0.0982378\pi\)
\(11\)	1.29208	−	1.62022i	0.389577	−	0.488514i	−0.547909	−	0.836538i	\(-0.684576\pi\)
							0.937485	+	0.348024i	\(0.113147\pi\)
\(13\)	−1.07269	−	0.0401372i	−0.297511	−	0.0111321i	−0.111781	−	0.993733i	\(-0.535655\pi\)
							−0.185730	+	0.982601i	\(0.559465\pi\)
\(17\)	0.132539	−	0.250776i	0.0321454	−	0.0608220i	−0.867520	−	0.497403i	\(-0.834287\pi\)
							0.899665	+	0.436581i	\(0.143811\pi\)
\(19\)	0.910724	−	0.137270i	0.208935	−	0.0314918i	−0.0437409	−	0.999043i	\(-0.513928\pi\)
							0.252675	+	0.967551i	\(0.418690\pi\)
\(23\)	−0.181560	+	0.416140i	−0.0378579	+	0.0867713i	−0.934464	−	0.356058i	\(-0.884121\pi\)
							0.896606	+	0.442829i	\(0.146025\pi\)
\(29\)	0.669707	+	8.93662i	0.124361	+	1.65949i	0.615299	+	0.788294i	\(0.289035\pi\)
							−0.490938	+	0.871195i	\(0.663346\pi\)
\(31\)	0.475039	−	0.323876i	0.0853195	−	0.0581699i	−0.519909	−	0.854222i	\(-0.674034\pi\)
							0.605228	+	0.796052i	\(0.293082\pi\)
\(37\)	6.20701	−	1.66316i	1.02043	−	0.273423i	0.290447	−	0.956891i	\(-0.406196\pi\)
							0.729980	+	0.683469i	\(0.239529\pi\)
\(41\)	2.14855	+	1.03469i	0.335548	+	0.161591i	0.594069	−	0.804414i	\(-0.297521\pi\)
							−0.258521	+	0.966006i	\(0.583235\pi\)
\(43\)	−6.54967	+	0.319196i	−0.998815	+	0.0486769i
							\(47\)	−2.57405	−	0.290026i	−0.375464	−	0.0423047i	−0.0777840	−	0.996970i	\(-0.524784\pi\)
−0.297681	+	0.954666i	\(0.596213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.18	✓	528
5.2	odd	4	inner	430.2.x.a.347.7	yes	528
43.29	odd	42	inner	430.2.x.a.373.7	yes	528
215.72	even	84	inner	430.2.x.a.287.18	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.18	✓	528	1.1	even	1	trivial
430.2.x.a.287.18	yes	528	215.72	even	84	inner
430.2.x.a.347.7	yes	528	5.2	odd	4	inner
430.2.x.a.373.7	yes	528	43.29	odd	42	inner