Embedded newform 430.2.r.a.27.18

• Label: 430.2.r.a.27.18
• 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			27.18
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.18

$q$-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 \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.18	✓	264	1.1	even	1	trivial
430.2.r.a.113.18	yes	264	5.3	odd	4	inner
430.2.r.a.137.18	yes	264	43.8	odd	14	inner
430.2.r.a.223.18	yes	264	215.8	even	28	inner