Embedded newform 430.2.r.a.323.15

• Label: 430.2.r.a.323.15
• Level: $430$
• Weight: $2$
• Character: 430.323
• Analytic conductor: $3.434$
• Analytic rank: $0$
• Dimension: $264$
• 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			323.15
Character	\(\chi\)	\(=\)	430.323
Dual form			430.2.r.a.217.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.330279 + 0.943883i) q^{2} +(-0.496329 + 1.41843i) q^{3} +(-0.781831 + 0.623490i) q^{4} +(0.656977 - 2.13738i) q^{5} -1.50276 q^{6} +(0.698925 + 0.698925i) q^{7} +(-0.846724 - 0.532032i) q^{8} +(0.579902 + 0.462457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(87\)	\(261\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.330279	+	0.943883i	0.233543	+	0.667426i
							\(3\)	−0.496329	+	1.41843i	−0.286556	+	0.818929i	0.707166	+	0.707048i	\(0.249973\pi\)
−0.993722	+	0.111881i	\(0.964312\pi\)
\(5\)	0.656977	−	2.13738i	0.293809	−	0.955864i
							\(7\)	0.698925	+	0.698925i	0.264169	+	0.264169i	0.826745	−	0.562576i	\(-0.190190\pi\)
−0.562576	+	0.826745i	\(0.690190\pi\)
\(11\)	−3.20933	+	4.02438i	−0.967651	+	1.21340i	0.00930574	+	0.999957i	\(0.497038\pi\)
							−0.976956	+	0.213439i	\(0.931534\pi\)
\(13\)	−3.36973	+	5.36290i	−0.934596	+	1.48740i	−0.0625125	+	0.998044i	\(0.519911\pi\)
							−0.872084	+	0.489357i	\(0.837232\pi\)
\(17\)	1.23104	+	1.95918i	0.298570	+	0.475172i	0.961993	−	0.273073i	\(-0.0880400\pi\)
							−0.663423	+	0.748244i	\(0.730897\pi\)
\(19\)	−0.339430	−	0.425631i	−0.0778705	−	0.0976466i	0.741370	−	0.671096i	\(-0.234176\pi\)
							−0.819241	+	0.573450i	\(0.805605\pi\)
\(23\)	9.27487	−	1.04503i	1.93394	−	0.217903i	0.941173	−	0.337926i	\(-0.109725\pi\)
							0.992771	+	0.120022i	\(0.0382967\pi\)
\(29\)	−5.54644	−	2.67103i	−1.02995	−	0.495997i	−0.158952	−	0.987286i	\(-0.550812\pi\)
							−0.870997	+	0.491289i	\(0.836526\pi\)
\(31\)	3.37017	+	1.62299i	0.605301	+	0.291497i	0.711322	−	0.702866i	\(-0.248097\pi\)
							−0.106021	+	0.994364i	\(0.533811\pi\)
\(37\)	−2.44596	−	2.44596i	−0.402113	−	0.402113i	0.476864	−	0.878977i	\(-0.341773\pi\)
							−0.878977	+	0.476864i	\(0.841773\pi\)
\(41\)	−0.867183	−	0.417613i	−0.135431	−	0.0652202i	0.364941	−	0.931031i	\(-0.381089\pi\)
							−0.500372	+	0.865810i	\(0.666803\pi\)
\(43\)	2.93077	+	5.86605i	0.446938	+	0.894565i
							\(47\)	9.67221	+	1.08980i	1.41084	+	0.158963i	0.784277	−	0.620411i	\(-0.213034\pi\)
0.626559	+	0.779374i	\(0.284463\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.323.15	yes	264
5.2	odd	4	inner	430.2.r.a.237.8	yes	264
43.2	odd	14	inner	430.2.r.a.303.8	yes	264
215.2	even	28	inner	430.2.r.a.217.15	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.217.15	✓	264	215.2	even	28	inner
430.2.r.a.237.8	yes	264	5.2	odd	4	inner
430.2.r.a.303.8	yes	264	43.2	odd	14	inner
430.2.r.a.323.15	yes	264	1.1	even	1	trivial