Embedded newform 430.2.r.a.27.4

• Label: 430.2.r.a.27.4
• 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			27.4
Character	\(\chi\)	\(=\)	430.27
Dual form			430.2.r.a.223.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.846724 - 0.532032i) q^{2} +(-1.25844 + 0.790731i) q^{3} +(0.433884 + 0.900969i) q^{4} +(1.73201 - 1.41427i) q^{5} +1.48625 q^{6} +(1.74142 - 1.74142i) q^{7} +(0.111964 - 0.993712i) q^{8} +(-0.343232 + 0.712728i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 8 q^{6}+O(q^{10}) \)

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
							\(3\)	−1.25844	+	0.790731i	−0.726562	+	0.456529i	−0.843857	−	0.536568i	\(-0.819720\pi\)
0.117295	+	0.993097i	\(0.462578\pi\)
\(5\)	1.73201	−	1.41427i	0.774577	−	0.632480i
							\(7\)	1.74142	−	1.74142i	0.658194	−	0.658194i	−0.296759	−	0.954952i	\(-0.595906\pi\)
0.954952	+	0.296759i	\(0.0959057\pi\)
\(11\)	−5.25211	−	2.52928i	−1.58357	−	0.762607i	−0.584752	−	0.811212i	\(-0.698808\pi\)
							−0.998818	+	0.0486047i	\(0.984523\pi\)
\(13\)	4.71581	+	0.531344i	1.30793	+	0.147368i	0.738266	−	0.674509i	\(-0.235645\pi\)
							0.569664	+	0.821878i	\(0.307073\pi\)
\(17\)	−1.52735	+	0.172091i	−0.370438	+	0.0417383i	−0.295221	−	0.955429i	\(-0.595393\pi\)
							−0.0752164	+	0.997167i	\(0.523965\pi\)
\(19\)	2.80005	−	1.34843i	0.642376	−	0.309352i	−0.0841886	−	0.996450i	\(-0.526830\pi\)
							0.726565	+	0.687098i	\(0.241116\pi\)
\(23\)	3.20397	−	1.12112i	0.668074	−	0.233769i	0.0251318	−	0.999684i	\(-0.491999\pi\)
							0.642942	+	0.765915i	\(0.277714\pi\)
\(29\)	0.538647	+	2.35997i	0.100024	+	0.438235i	0.999997	+	0.00228967i	\(0.000728824\pi\)
							−0.899973	+	0.435946i	\(0.856414\pi\)
\(31\)	−2.34816	−	10.2880i	−0.421743	−	1.84778i	−0.522185	−	0.852832i	\(-0.674883\pi\)
							0.100442	−	0.994943i	\(-0.467974\pi\)
\(37\)	3.50941	−	3.50941i	0.576943	−	0.576943i	−0.357117	−	0.934060i	\(-0.616240\pi\)
							0.934060	+	0.357117i	\(0.116240\pi\)
\(41\)	−0.581212	−	2.54646i	−0.0907701	−	0.397690i	0.909049	−	0.416688i	\(-0.136809\pi\)
							−0.999820	+	0.0189986i	\(0.993952\pi\)
\(43\)	5.31220	−	3.84454i	0.810104	−	0.586287i
							\(47\)	7.91513	+	2.76963i	1.15454	+	0.403991i	0.838579	−	0.544780i	\(-0.183387\pi\)
0.315962	+	0.948772i	\(0.397673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	430.2.r.a.27.4	✓	264
5.3	odd	4	inner	430.2.r.a.113.4	yes	264
43.8	odd	14	inner	430.2.r.a.137.4	yes	264
215.8	even	28	inner	430.2.r.a.223.4	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
430.2.r.a.27.4	✓	264	1.1	even	1	trivial
430.2.r.a.113.4	yes	264	5.3	odd	4	inner
430.2.r.a.137.4	yes	264	43.8	odd	14	inner
430.2.r.a.223.4	yes	264	215.8	even	28	inner