Embedded newform 430.2.r.a.27.20

• Label: 430.2.r.a.27.20
• Level: $430$
• Weight: $2$
• Character: 430.27
• 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			27.20
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.846724 + 0.532032i) q^{2} +(1.04232 - 0.654933i) q^{3} +(0.433884 + 0.900969i) q^{4} +(-0.0867089 + 2.23439i) q^{5} +1.23100 q^{6} +(2.75433 - 2.75433i) q^{7} +(-0.111964 + 0.993712i) q^{8} +(-0.644158 + 1.33761i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{14}\right)\)

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
							\(3\)	1.04232	−	0.654933i	0.601783	−	0.378126i	−0.196414	−	0.980521i	\(-0.562930\pi\)
0.798198	+	0.602395i	\(0.205787\pi\)
\(5\)	−0.0867089	+	2.23439i	−0.0387774	+	0.999248i
							\(7\)	2.75433	−	2.75433i	1.04104	−	1.04104i	0.0419179	−	0.999121i	\(-0.486653\pi\)
0.999121	−	0.0419179i	\(-0.0133468\pi\)
\(11\)	2.66926	+	1.28545i	0.804813	+	0.387578i	0.790609	−	0.612322i	\(-0.209764\pi\)
							0.0142043	+	0.999899i	\(0.495478\pi\)
\(13\)	−5.90084	−	0.664865i	−1.63660	−	0.184400i	−0.754776	−	0.655982i	\(-0.772255\pi\)
							−0.881822	+	0.471582i	\(0.843683\pi\)
\(17\)	1.33078	−	0.149943i	0.322763	−	0.0363666i	0.0509032	−	0.998704i	\(-0.483790\pi\)
							0.271859	+	0.962337i	\(0.412361\pi\)
\(19\)	6.87043	−	3.30863i	1.57619	−	0.759051i	0.577818	−	0.816166i	\(-0.303904\pi\)
							0.998368	+	0.0571148i	\(0.0181901\pi\)
\(23\)	−3.85558	+	1.34913i	−0.803944	+	0.281312i	−0.700803	−	0.713354i	\(-0.747175\pi\)
							−0.103141	+	0.994667i	\(0.532889\pi\)
\(29\)	−0.296286	−	1.29812i	−0.0550190	−	0.241054i	0.939940	−	0.341341i	\(-0.110881\pi\)
							−0.994959	+	0.100287i	\(0.968024\pi\)
\(31\)	−0.943183	−	4.13236i	−0.169401	−	0.742193i	−0.986239	−	0.165326i	\(-0.947132\pi\)
							0.816838	−	0.576867i	\(-0.195725\pi\)
\(37\)	−1.54397	+	1.54397i	−0.253826	+	0.253826i	−0.822537	−	0.568711i	\(-0.807442\pi\)
							0.568711	+	0.822537i	\(0.307442\pi\)
\(41\)	0.434837	+	1.90514i	0.0679101	+	0.297533i	0.997466	−	0.0711483i	\(-0.0226664\pi\)
							−0.929556	+	0.368682i	\(0.879809\pi\)
\(43\)	−0.471137	−	6.54049i	−0.0718477	−	0.997416i
							\(47\)	−0.895465	−	0.313337i	−0.130617	−	0.0457049i	0.264179	−	0.964474i	\(-0.414899\pi\)
−0.394796	+	0.918769i	\(0.629185\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.27.20	✓	264
5.3	odd	4	inner	430.2.r.a.113.20	yes	264
43.8	odd	14	inner	430.2.r.a.137.20	yes	264
215.8	even	28	inner	430.2.r.a.223.20	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.20	✓	264	1.1	even	1	trivial
430.2.r.a.113.20	yes	264	5.3	odd	4	inner
430.2.r.a.137.20	yes	264	43.8	odd	14	inner
430.2.r.a.223.20	yes	264	215.8	even	28	inner