Embedded newform 150.3.j.a.11.4

• Label: 150.3.j.a.11.4
• Level: $150$
• Weight: $3$
• Character: 150.11
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(20\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			11.4
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 - 1.14412i) q^{2} +(-1.90427 + 2.31814i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(-4.93889 + 0.779331i) q^{5} +(4.23516 + 0.251761i) q^{6} +8.16986 q^{7} +(2.68999 - 0.874032i) q^{8} +(-1.74750 - 8.82872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.831254	−	1.14412i	−0.415627	−	0.572061i
							\(3\)	−1.90427	+	2.31814i	−0.634757	+	0.772712i
\(5\)	−4.93889	+	0.779331i	−0.987778	+	0.155866i
							\(7\)	8.16986			1.16712			0.583562	−	0.812069i	\(-0.301659\pi\)
0.583562	+	0.812069i	\(0.301659\pi\)
\(11\)	−6.71524	−	9.24273i	−0.610476	−	0.840249i	0.386140	−	0.922440i	\(-0.373808\pi\)
							−0.996617	+	0.0821915i	\(0.973808\pi\)
\(13\)	−16.0096	−	11.6317i	−1.23151	−	0.894745i	−0.234508	−	0.972114i	\(-0.575348\pi\)
							−0.997002	+	0.0773696i	\(0.975348\pi\)
\(17\)	14.3824	−	4.67311i	0.846021	−	0.274889i	0.146243	−	0.989249i	\(-0.453282\pi\)
							0.699779	+	0.714360i	\(0.253282\pi\)
\(19\)	−4.39373	−	13.5225i	−0.231249	−	0.711711i	−0.997597	−	0.0692850i	\(-0.977928\pi\)
							0.766348	−	0.642426i	\(-0.222072\pi\)
\(23\)	1.26018	+	1.73449i	0.0547903	+	0.0754124i	0.835533	−	0.549440i	\(-0.185159\pi\)
							−0.780743	+	0.624853i	\(0.785159\pi\)
\(29\)	−44.8797	−	14.5823i	−1.54758	−	0.502838i	−0.594122	−	0.804375i	\(-0.702500\pi\)
							−0.953455	+	0.301537i	\(0.902500\pi\)
\(31\)	−12.3974	−	38.1554i	−0.399918	−	1.23082i	−0.925065	−	0.379808i	\(-0.875990\pi\)
							0.525148	−	0.851011i	\(-0.324010\pi\)
\(37\)	42.2525	+	30.6983i	1.14196	+	0.829682i	0.987391	−	0.158300i	\(-0.0506012\pi\)
							0.154569	+	0.987982i	\(0.450601\pi\)
\(41\)	−12.3449	+	16.9912i	−0.301094	+	0.414421i	−0.932578	−	0.360968i	\(-0.882446\pi\)
							0.631484	+	0.775389i	\(0.282446\pi\)
\(43\)	−22.1632			−0.515423			−0.257712	−	0.966222i	\(-0.582968\pi\)
							−0.257712	+	0.966222i	\(0.582968\pi\)
\(47\)	−35.0106	−	11.3756i	−0.744906	−	0.242035i	−0.0881183	−	0.996110i	\(-0.528085\pi\)
							−0.656788	+	0.754075i	\(0.728085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.4	✓	80
3.2	odd	2	inner	150.3.j.a.11.16	yes	80
25.16	even	5	inner	150.3.j.a.41.16	yes	80
75.41	odd	10	inner	150.3.j.a.41.4	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.4	✓	80	1.1	even	1	trivial
150.3.j.a.11.16	yes	80	3.2	odd	2	inner
150.3.j.a.41.4	yes	80	75.41	odd	10	inner
150.3.j.a.41.16	yes	80	25.16	even	5	inner