Embedded newform 430.2.r.a.223.5

• Label: 430.2.r.a.223.5
• Level: $430$
• Weight: $2$
• Character: 430.223
• 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.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			223.5
Character	\(\chi\)	\(=\)	430.223
Dual form			430.2.r.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.846724 + 0.532032i) q^{2} +(-0.0459382 - 0.0288649i) q^{3} +(0.433884 - 0.900969i) q^{4} +(-1.90179 + 1.17609i) q^{5} +0.0542540 q^{6} +(-0.499604 - 0.499604i) q^{7} +(0.111964 + 0.993712i) q^{8} +(-1.30037 - 2.70025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.846724	+	0.532032i	−0.598724	+	0.376203i
							\(3\)	−0.0459382	−	0.0288649i	−0.0265224	−	0.0166651i	0.518706	−	0.854953i	\(-0.326414\pi\)
−0.545228	+	0.838288i	\(0.683557\pi\)
\(5\)	−1.90179	+	1.17609i	−0.850508	+	0.525962i
							\(7\)	−0.499604	−	0.499604i	−0.188833	−	0.188833i	0.606359	−	0.795191i	\(-0.292630\pi\)
−0.795191	+	0.606359i	\(0.792630\pi\)
\(11\)	1.98628	−	0.956540i	0.598885	−	0.288408i	−0.109777	−	0.993956i	\(-0.535014\pi\)
							0.708661	+	0.705549i	\(0.249299\pi\)
\(13\)	4.43413	−	0.499607i	1.22981	−	0.138566i	0.526950	−	0.849896i	\(-0.323336\pi\)
							0.702858	+	0.711330i	\(0.251907\pi\)
\(17\)	2.69020	+	0.303112i	0.652469	+	0.0735156i	0.431993	−	0.901877i	\(-0.357811\pi\)
							0.220476	+	0.975392i	\(0.429239\pi\)
\(19\)	−0.517113	−	0.249028i	−0.118634	−	0.0571310i	0.373626	−	0.927580i	\(-0.378115\pi\)
							−0.492259	+	0.870449i	\(0.663829\pi\)
\(23\)	5.75294	+	2.01304i	1.19957	+	0.419748i	0.854809	−	0.518942i	\(-0.173674\pi\)
							0.344761	+	0.938690i	\(0.387960\pi\)
\(29\)	0.0156851	−	0.0687207i	0.00291264	−	0.0127611i	−0.973451	−	0.228897i	\(-0.926488\pi\)
							0.976363	+	0.216136i	\(0.0693453\pi\)
\(31\)	1.84639	−	8.08958i	0.331622	−	1.45293i	−0.484367	−	0.874865i	\(-0.660950\pi\)
							0.815989	−	0.578067i	\(-0.196193\pi\)
\(37\)	−1.14650	−	1.14650i	−0.188484	−	0.188484i	0.606557	−	0.795040i	\(-0.292550\pi\)
							−0.795040	+	0.606557i	\(0.792550\pi\)
\(41\)	−0.823879	+	3.60965i	−0.128668	+	0.563732i	0.868959	+	0.494884i	\(0.164790\pi\)
							−0.997627	+	0.0688481i	\(0.978068\pi\)
\(43\)	6.51895	+	0.709388i	0.994131	+	0.108181i
							\(47\)	2.75594	−	0.964344i	0.401995	−	0.140664i	−0.121706	−	0.992566i	\(-0.538837\pi\)
0.523701	+	0.851902i	\(0.324551\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.223.5	yes	264
5.2	odd	4	inner	430.2.r.a.137.5	yes	264
43.27	odd	14	inner	430.2.r.a.113.5	yes	264
215.27	even	28	inner	430.2.r.a.27.5	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.5	✓	264	215.27	even	28	inner
430.2.r.a.113.5	yes	264	43.27	odd	14	inner
430.2.r.a.137.5	yes	264	5.2	odd	4	inner
430.2.r.a.223.5	yes	264	1.1	even	1	trivial