Embedded newform 430.2.r.a.223.18

• Label: 430.2.r.a.223.18
• Level: $430$
• Weight: $2$
• Character: 430.223
• 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			223.18
Character	\(\chi\)	\(=\)	430.223
Dual form			430.2.r.a.27.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.846724 - 0.532032i) q^{2} +(0.919405 + 0.577701i) q^{3} +(0.433884 - 0.900969i) q^{4} +(-1.69116 - 1.46286i) q^{5} +1.08584 q^{6} +(-2.01843 - 2.01843i) q^{7} +(-0.111964 - 0.993712i) q^{8} +(-0.790083 - 1.64062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	0.919405	+	0.577701i	0.530819	+	0.333536i	0.770629	−	0.637284i	\(-0.219942\pi\)
−0.239810	+	0.970820i	\(0.577085\pi\)
\(5\)	−1.69116	−	1.46286i	−0.756311	−	0.654213i
							\(7\)	−2.01843	−	2.01843i	−0.762895	−	0.762895i	0.213950	−	0.976845i	\(-0.431367\pi\)
−0.976845	+	0.213950i	\(0.931367\pi\)
\(11\)	2.48015	−	1.19438i	0.747794	−	0.360119i	−0.0208607	−	0.999782i	\(-0.506641\pi\)
							0.768655	+	0.639664i	\(0.220926\pi\)
\(13\)	3.64645	−	0.410856i	1.01134	−	0.113951i	0.409288	−	0.912405i	\(-0.365777\pi\)
							0.602055	+	0.798454i	\(0.294349\pi\)
\(17\)	−1.55803	−	0.175548i	−0.377878	−	0.0425767i	−0.0790172	−	0.996873i	\(-0.525178\pi\)
							−0.298861	+	0.954297i	\(0.596607\pi\)
\(19\)	−2.12004	−	1.02096i	−0.486371	−	0.234224i	0.174594	−	0.984640i	\(-0.444139\pi\)
							−0.660965	+	0.750417i	\(0.729853\pi\)
\(23\)	6.32004	+	2.21148i	1.31782	+	0.461125i	0.895395	−	0.445272i	\(-0.146893\pi\)
							0.422425	+	0.906398i	\(0.361179\pi\)
\(29\)	−0.632790	+	2.77244i	−0.117506	+	0.514828i	0.881578	+	0.472039i	\(0.156482\pi\)
							−0.999084	+	0.0427897i	\(0.986375\pi\)
\(31\)	−0.865388	+	3.79151i	−0.155428	+	0.680976i	0.835824	+	0.548997i	\(0.184990\pi\)
							−0.991253	+	0.131979i	\(0.957867\pi\)
\(37\)	7.67346	+	7.67346i	1.26151	+	1.26151i	0.950362	+	0.311146i	\(0.100713\pi\)
							0.311146	+	0.950362i	\(0.399287\pi\)
\(41\)	0.207329	−	0.908366i	0.0323793	−	0.141863i	−0.956155	−	0.292863i	\(-0.905392\pi\)
							0.988534	+	0.151000i	\(0.0482493\pi\)
\(43\)	−6.38590	−	1.49006i	−0.973841	−	0.227232i
							\(47\)	0.375718	−	0.131470i	0.0548041	−	0.0191768i	−0.302738	−	0.953074i	\(-0.597901\pi\)
0.357542	+	0.933897i	\(0.383615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.223.18	yes	264
5.2	odd	4	inner	430.2.r.a.137.18	yes	264
43.27	odd	14	inner	430.2.r.a.113.18	yes	264
215.27	even	28	inner	430.2.r.a.27.18	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.18	✓	264	215.27	even	28	inner
430.2.r.a.113.18	yes	264	43.27	odd	14	inner
430.2.r.a.137.18	yes	264	5.2	odd	4	inner
430.2.r.a.223.18	yes	264	1.1	even	1	trivial