Embedded newform 154.3.p.a.39.10

• Label: 154.3.p.a.39.10
• Level: $154$
• Weight: $3$
• Character: 154.39
• Analytic conductor: $4.196$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 154 = 2 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	154.p (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19619607115\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(16\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			39.10
Character	\(\chi\)	\(=\)	154.39
Dual form			154.3.p.a.79.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05097 + 0.946294i) q^{2} +(-3.54170 + 1.57686i) q^{3} +(0.209057 + 1.98904i) q^{4} +(4.72709 - 1.00477i) q^{5} +(-5.21438 - 1.69425i) q^{6} +(-1.14894 + 6.90507i) q^{7} +(-1.66251 + 2.28825i) q^{8} +(4.03493 - 4.48125i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q - 32 q^{4} - 4 q^{5} + 30 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).

\(n\)	\(45\)	\(57\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05097	+	0.946294i	0.525483	+	0.473147i
							\(3\)	−3.54170	+	1.57686i	−1.18057	+	0.525622i	−0.900710	−	0.434421i	\(-0.856953\pi\)
−0.279855	+	0.960042i	\(0.590286\pi\)
\(5\)	4.72709	−	1.00477i	0.945419	−	0.200955i	0.290689	−	0.956818i	\(-0.406116\pi\)
							0.654730	+	0.755863i	\(0.272782\pi\)
\(7\)	−1.14894	+	6.90507i	−0.164135	+	0.986438i
							\(11\)	−9.76749	+	5.05927i	−0.887953	+	0.459933i
\(13\)	−17.2578	+	5.60740i	−1.32752	+	0.431338i								−0.885072	−	0.465455i	\(-0.845891\pi\)
							−0.442451	+	0.896793i	\(0.645891\pi\)
\(17\)	15.0568	−	13.5572i	0.885694	−	0.797482i	−0.0944997	−	0.995525i	\(-0.530125\pi\)
							0.980193	+	0.198043i	\(0.0634585\pi\)
\(19\)	−9.92013	−	1.04265i	−0.522112	−	0.0548762i	−0.160193	−	0.987086i	\(-0.551212\pi\)
							−0.361919	+	0.932209i	\(0.617878\pi\)
\(23\)	19.9160	+	34.4956i	0.865915	+	1.49981i	0.866136	+	0.499808i	\(0.166596\pi\)
							−0.000221753		1.00000i	\(0.500071\pi\)
\(29\)	10.9411	+	15.0591i	0.377278	+	0.519278i	0.954861	−	0.297054i	\(-0.0960041\pi\)
							−0.577583	+	0.816332i	\(0.696004\pi\)
\(31\)	22.0366	+	4.68403i	0.710860	+	0.151098i	0.549130	−	0.835737i	\(-0.314959\pi\)
							0.161730	+	0.986835i	\(0.448293\pi\)
\(37\)	63.9321	+	28.4644i	1.72789	+	0.769308i	0.996144	+	0.0877362i	\(0.0279633\pi\)
							0.731751	+	0.681572i	\(0.238703\pi\)
\(41\)	33.4602	−	46.0540i	0.816101	−	1.12327i	−0.174252	−	0.984701i	\(-0.555751\pi\)
							0.990353	−	0.138566i	\(-0.0442493\pi\)
\(43\)			15.1973i			0.353425i	0.984263	+	0.176712i	\(0.0565463\pi\)
							−0.984263	+	0.176712i	\(0.943454\pi\)
\(47\)	0.0791145	−	0.752725i	0.00168329	−	0.0160154i	−0.993648	−	0.112533i	\(-0.964104\pi\)
							0.995331	+	0.0965174i	\(0.0307703\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	154.3.p.a.39.10	✓	128
7.2	even	3	inner	154.3.p.a.149.15	yes	128
11.2	odd	10	inner	154.3.p.a.123.15	yes	128
77.2	odd	30	inner	154.3.p.a.79.10	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.3.p.a.39.10	✓	128	1.1	even	1	trivial
154.3.p.a.79.10	yes	128	77.2	odd	30	inner
154.3.p.a.123.15	yes	128	11.2	odd	10	inner
154.3.p.a.149.15	yes	128	7.2	even	3	inner