Embedded newform 430.2.x.a.3.7

• Label: 430.2.x.a.3.7
• 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			3.7
Character	\(\chi\)	\(=\)	430.3
Dual form			430.2.x.a.287.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.330279 - 0.943883i) q^{2} +(0.658429 + 0.765108i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(-0.824716 - 2.07842i) q^{5} +(0.504708 - 0.874179i) q^{6} +(-1.09920 + 4.10225i) q^{7} +(0.846724 + 0.532032i) q^{8} +(0.295265 - 1.95895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 4 q^{6} - 12 q^{7}+O(q^{10}) \)

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
							\(3\)	0.658429	+	0.765108i	0.380144	+	0.441736i	0.915150	−	0.403113i	\(-0.132072\pi\)
−0.535006	+	0.844848i	\(0.679691\pi\)
\(5\)	−0.824716	−	2.07842i	−0.368824	−	0.929499i
							\(7\)	−1.09920	+	4.10225i	−0.415457	+	1.55051i	0.368462	+	0.929643i	\(0.379884\pi\)
−0.783919	+	0.620863i	\(0.786782\pi\)
\(11\)	2.78882	−	3.49707i	0.840861	−	1.05441i	−0.156906	−	0.987614i	\(-0.550152\pi\)
							0.997767	−	0.0667925i	\(-0.0212766\pi\)
\(13\)	2.68965	+	0.100640i	0.745976	+	0.0279124i	0.407677	−	0.913126i	\(-0.366339\pi\)
							0.338299	+	0.941039i	\(0.390149\pi\)
\(17\)	3.22477	−	6.10157i	0.782122	−	1.47985i	−0.0928529	−	0.995680i	\(-0.529599\pi\)
							0.874975	−	0.484168i	\(-0.160878\pi\)
\(19\)	7.38176	−	1.11262i	1.69349	−	0.255253i	0.769771	−	0.638320i	\(-0.220370\pi\)
							0.923720	+	0.383067i	\(0.125132\pi\)
\(23\)	−2.03016	+	4.65317i	−0.423317	+	0.970253i	0.565991	+	0.824411i	\(0.308494\pi\)
							−0.989308	+	0.145841i	\(0.953411\pi\)
\(29\)	−0.340558	−	4.54443i	−0.0632400	−	0.843879i	−0.935159	−	0.354228i	\(-0.884744\pi\)
							0.871919	−	0.489650i	\(-0.162876\pi\)
\(31\)	−6.62748	+	4.51854i	−1.19033	+	0.811554i	−0.985901	−	0.167328i	\(-0.946486\pi\)
							−0.204429	+	0.978881i	\(0.565534\pi\)
\(37\)	3.97050	−	1.06389i	0.652746	−	0.174903i	0.0827757	−	0.996568i	\(-0.473621\pi\)
							0.569970	+	0.821666i	\(0.306955\pi\)
\(41\)	−0.634590	−	0.305602i	−0.0991062	−	0.0477271i	0.383674	−	0.923469i	\(-0.374659\pi\)
							−0.482780	+	0.875742i	\(0.660373\pi\)
\(43\)	4.42425	+	4.84004i	0.674692	+	0.738099i
							\(47\)	−1.44762	−	0.163107i	−0.211157	−	0.0237916i	0.00575246	−	0.999983i	\(-0.498169\pi\)
−0.216909	+	0.976192i	\(0.569597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.x.a.3.7	✓	528
5.2	odd	4	inner	430.2.x.a.347.18	yes	528
43.29	odd	42	inner	430.2.x.a.373.18	yes	528
215.72	even	84	inner	430.2.x.a.287.7	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.x.a.3.7	✓	528	1.1	even	1	trivial
430.2.x.a.287.7	yes	528	215.72	even	84	inner
430.2.x.a.347.18	yes	528	5.2	odd	4	inner
430.2.x.a.373.18	yes	528	43.29	odd	42	inner