Embedded newform 150.3.j.a.11.5

• Label: 150.3.j.a.11.5
• 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			11.5
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.831254 - 1.14412i) q^{2} +(-0.513013 + 2.95581i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(2.36498 - 4.40532i) q^{5} +(3.80825 - 1.87008i) q^{6} +1.96623 q^{7} +(2.68999 - 0.874032i) q^{8} +(-8.47364 - 3.03274i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.513013	+	2.95581i	−0.171004	+	0.985270i
\(5\)	2.36498	−	4.40532i	0.472995	−	0.881065i
							\(7\)	1.96623			0.280889			0.140445	−	0.990089i	\(-0.455147\pi\)
0.140445	+	0.990089i	\(0.455147\pi\)
\(11\)	8.25152	+	11.3572i	0.750138	+	1.03248i	0.997971	+	0.0636741i	\(0.0202818\pi\)
							−0.247832	+	0.968803i	\(0.579718\pi\)
\(13\)	13.4390	+	9.76402i	1.03377	+	0.751078i	0.969060	−	0.246826i	\(-0.0793875\pi\)
							0.0647108	+	0.997904i	\(0.479388\pi\)
\(17\)	23.1929	−	7.53582i	1.36429	−	0.443284i	0.466815	−	0.884355i	\(-0.345402\pi\)
							0.897472	+	0.441072i	\(0.145402\pi\)
\(19\)	8.24042	+	25.3614i	0.433706	+	1.33481i	0.894407	+	0.447255i	\(0.147598\pi\)
							−0.460701	+	0.887556i	\(0.652402\pi\)
\(23\)	−9.75557	−	13.4274i	−0.424155	−	0.583800i	0.542444	−	0.840092i	\(-0.317499\pi\)
							−0.966599	+	0.256292i	\(0.917499\pi\)
\(29\)	−11.7344	−	3.81274i	−0.404635	−	0.131474i	0.0996263	−	0.995025i	\(-0.468235\pi\)
							−0.504262	+	0.863551i	\(0.668235\pi\)
\(31\)	8.53426	+	26.2657i	0.275299	+	0.847282i	0.989140	+	0.146975i	\(0.0469537\pi\)
							−0.713842	+	0.700307i	\(0.753046\pi\)
\(37\)	−8.68504	−	6.31005i	−0.234731	−	0.170542i	0.464202	−	0.885729i	\(-0.346341\pi\)
							−0.698933	+	0.715188i	\(0.746341\pi\)
\(41\)	15.4757	−	21.3004i	0.377455	−	0.519523i	−0.577453	−	0.816424i	\(-0.695953\pi\)
							0.954908	+	0.296901i	\(0.0959533\pi\)
\(43\)	66.0299			1.53558			0.767790	−	0.640701i	\(-0.221356\pi\)
							0.767790	+	0.640701i	\(0.221356\pi\)
\(47\)	−43.5204	−	14.1406i	−0.925966	−	0.300864i	−0.193054	−	0.981188i	\(-0.561839\pi\)
							−0.732912	+	0.680324i	\(0.761839\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.5	✓	80
3.2	odd	2	inner	150.3.j.a.11.15	yes	80
25.16	even	5	inner	150.3.j.a.41.15	yes	80
75.41	odd	10	inner	150.3.j.a.41.5	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.5	✓	80	1.1	even	1	trivial
150.3.j.a.11.15	yes	80	3.2	odd	2	inner
150.3.j.a.41.5	yes	80	75.41	odd	10	inner
150.3.j.a.41.15	yes	80	25.16	even	5	inner