Embedded newform 150.3.j.a.11.3

• Label: 150.3.j.a.11.3
• 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			11.3
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 - 1.14412i) q^{2} +(-2.21254 - 2.02600i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(1.16748 + 4.86179i) q^{5} +(-0.478809 + 4.21554i) q^{6} +6.03242 q^{7} +(2.68999 - 0.874032i) q^{8} +(0.790661 + 8.96520i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.21254	−	2.02600i	−0.737513	−	0.675333i
\(5\)	1.16748	+	4.86179i	0.233496	+	0.972358i
							\(7\)	6.03242			0.861775			0.430887	−	0.902406i	\(-0.358201\pi\)
0.430887	+	0.902406i	\(0.358201\pi\)
\(11\)	−3.18447	−	4.38305i	−0.289497	−	0.398459i	0.639353	−	0.768913i	\(-0.279202\pi\)
							−0.928851	+	0.370454i	\(0.879202\pi\)
\(13\)	10.2465	+	7.44455i	0.788196	+	0.572658i	0.907428	−	0.420208i	\(-0.138043\pi\)
							−0.119231	+	0.992866i	\(0.538043\pi\)
\(17\)	29.2114	−	9.49135i	1.71832	−	0.558314i	0.726633	−	0.687026i	\(-0.241084\pi\)
							0.991682	+	0.128711i	\(0.0410841\pi\)
\(19\)	−5.28636	−	16.2697i	−0.278229	−	0.856302i	−0.988347	−	0.152219i	\(-0.951358\pi\)
							0.710117	−	0.704083i	\(-0.248642\pi\)
\(23\)	4.21551	+	5.80215i	0.183283	+	0.252268i	0.890765	−	0.454464i	\(-0.150169\pi\)
							−0.707482	+	0.706731i	\(0.750169\pi\)
\(29\)	37.6522	+	12.2340i	1.29835	+	0.421860i	0.875008	−	0.484108i	\(-0.160856\pi\)
							0.423345	+	0.905969i	\(0.360856\pi\)
\(31\)	10.1306	+	31.1789i	0.326795	+	1.00577i	0.970624	+	0.240601i	\(0.0773446\pi\)
							−0.643829	+	0.765169i	\(0.722655\pi\)
\(37\)	10.6591	+	7.74430i	0.288084	+	0.209305i	0.722436	−	0.691438i	\(-0.243022\pi\)
							−0.434352	+	0.900743i	\(0.643022\pi\)
\(41\)	−27.3403	+	37.6306i	−0.666836	+	0.917821i	−0.999683	−	0.0251600i	\(-0.991990\pi\)
							0.332848	+	0.942981i	\(0.391990\pi\)
\(43\)	2.54995			0.0593013			0.0296506	−	0.999560i	\(-0.490561\pi\)
							0.0296506	+	0.999560i	\(0.490561\pi\)
\(47\)	61.8997	+	20.1124i	1.31702	+	0.427924i	0.881469	−	0.472242i	\(-0.156555\pi\)
							0.435547	+	0.900166i	\(0.356555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.3	✓	80
3.2	odd	2	inner	150.3.j.a.11.20	yes	80
25.16	even	5	inner	150.3.j.a.41.20	yes	80
75.41	odd	10	inner	150.3.j.a.41.3	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.3	✓	80	1.1	even	1	trivial
150.3.j.a.11.20	yes	80	3.2	odd	2	inner
150.3.j.a.41.3	yes	80	75.41	odd	10	inner
150.3.j.a.41.20	yes	80	25.16	even	5	inner