Embedded newform 150.3.j.a.11.9

• Label: 150.3.j.a.11.9
• 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			11.9
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 - 1.14412i) q^{2} +(2.78863 + 1.10613i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(-4.36319 + 2.44184i) q^{5} +(-1.05251 - 4.11002i) q^{6} +4.66363 q^{7} +(2.68999 - 0.874032i) q^{8} +(6.55293 + 6.16920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	2.78863	+	1.10613i	0.929544	+	0.368711i
\(5\)	−4.36319	+	2.44184i	−0.872638	+	0.488367i
							\(7\)	4.66363			0.666233			0.333116	−	0.942886i	\(-0.391900\pi\)
0.333116	+	0.942886i	\(0.391900\pi\)
\(11\)	2.09360	+	2.88160i	0.190328	+	0.261964i	0.893507	−	0.449049i	\(-0.148237\pi\)
							−0.703180	+	0.711012i	\(0.748237\pi\)
\(13\)	18.2244	+	13.2408i	1.40188	+	1.01852i	0.994441	+	0.105291i	\(0.0335775\pi\)
							0.407437	+	0.913233i	\(0.366423\pi\)
\(17\)	−11.5884	+	3.76531i	−0.681672	+	0.221489i	−0.629327	−	0.777140i	\(-0.716669\pi\)
							−0.0523448	+	0.998629i	\(0.516669\pi\)
\(19\)	3.82708	+	11.7786i	0.201425	+	0.619924i	0.999841	+	0.0178173i	\(0.00567174\pi\)
							−0.798416	+	0.602106i	\(0.794328\pi\)
\(23\)	−7.54792	−	10.3888i	−0.328170	−	0.451688i	0.612769	−	0.790262i	\(-0.290055\pi\)
							−0.940940	+	0.338574i	\(0.890055\pi\)
\(29\)	26.8199	+	8.71430i	0.924823	+	0.300493i	0.732444	−	0.680828i	\(-0.238380\pi\)
							0.192379	+	0.981321i	\(0.438380\pi\)
\(31\)	−16.6071	−	51.1114i	−0.535713	−	1.64876i	−0.742103	−	0.670285i	\(-0.766172\pi\)
							0.206390	−	0.978470i	\(-0.433828\pi\)
\(37\)	−12.3397	−	8.96535i	−0.333506	−	0.242307i	0.408411	−	0.912798i	\(-0.366083\pi\)
							−0.741917	+	0.670492i	\(0.766083\pi\)
\(41\)	14.9764	−	20.6133i	0.365279	−	0.502763i	−0.586331	−	0.810071i	\(-0.699428\pi\)
							0.951610	+	0.307308i	\(0.0994283\pi\)
\(43\)	−31.9780			−0.743675			−0.371837	−	0.928298i	\(-0.621272\pi\)
							−0.371837	+	0.928298i	\(0.621272\pi\)
\(47\)	−18.9392	−	6.15370i	−0.402961	−	0.130930i	0.100523	−	0.994935i	\(-0.467948\pi\)
							−0.503483	+	0.864005i	\(0.667948\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.9	✓	80
3.2	odd	2	inner	150.3.j.a.11.11	yes	80
25.16	even	5	inner	150.3.j.a.41.11	yes	80
75.41	odd	10	inner	150.3.j.a.41.9	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.9	✓	80	1.1	even	1	trivial
150.3.j.a.11.11	yes	80	3.2	odd	2	inner
150.3.j.a.41.9	yes	80	75.41	odd	10	inner
150.3.j.a.41.11	yes	80	25.16	even	5	inner