Embedded newform 150.3.j.a.11.13

• Label: 150.3.j.a.11.13
• 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			11.13
Character	\(\chi\)	\(=\)	150.11
Dual form			150.3.j.a.41.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.831254 + 1.14412i) q^{2} +(-1.69605 - 2.47455i) q^{3} +(-0.618034 + 1.90211i) q^{4} +(1.09537 + 4.87854i) q^{5} +(1.42134 - 3.99747i) q^{6} -13.2558 q^{7} +(-2.68999 + 0.874032i) q^{8} +(-3.24680 + 8.39394i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 40 q^{4} - 8 q^{7} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.69605	−	2.47455i	−0.565351	−	0.824850i
\(5\)	1.09537	+	4.87854i	0.219075	+	0.975708i
							\(7\)	−13.2558			−1.89368			−0.946840	−	0.321704i	\(-0.895744\pi\)
−0.946840	+	0.321704i	\(0.895744\pi\)
\(11\)	9.38221	+	12.9135i	0.852928	+	1.17395i	0.983210	+	0.182478i	\(0.0584118\pi\)
							−0.130282	+	0.991477i	\(0.541588\pi\)
\(13\)	7.27732	+	5.28729i	0.559794	+	0.406714i	0.831384	−	0.555699i	\(-0.187549\pi\)
							−0.271589	+	0.962413i	\(0.587549\pi\)
\(17\)	−21.6027	+	7.01914i	−1.27075	+	0.412891i	−0.865312	−	0.501234i	\(-0.832880\pi\)
							−0.405434	+	0.914124i	\(0.632880\pi\)
\(19\)	−3.03572	−	9.34298i	−0.159775	−	0.491736i	0.838839	−	0.544380i	\(-0.183235\pi\)
							−0.998613	+	0.0526442i	\(0.983235\pi\)
\(23\)	−8.66379	−	11.9247i	−0.376686	−	0.518464i	0.578016	−	0.816025i	\(-0.303827\pi\)
							−0.954703	+	0.297561i	\(0.903827\pi\)
\(29\)	18.8313	+	6.11867i	0.649356	+	0.210989i	0.615130	−	0.788426i	\(-0.289103\pi\)
							0.0342257	+	0.999414i	\(0.489103\pi\)
\(31\)	−5.76110	−	17.7309i	−0.185842	−	0.571963i	0.814120	−	0.580697i	\(-0.197220\pi\)
							−0.999962	+	0.00873392i	\(0.997220\pi\)
\(37\)	36.6724	+	26.6440i	0.991145	+	0.720109i	0.960172	−	0.279411i	\(-0.0901392\pi\)
							0.0309738	+	0.999520i	\(0.490139\pi\)
\(41\)	−4.68042	+	6.44204i	−0.114157	+	0.157123i	−0.862272	−	0.506446i	\(-0.830959\pi\)
							0.748115	+	0.663569i	\(0.230959\pi\)
\(43\)	44.3127			1.03053			0.515264	−	0.857031i	\(-0.327694\pi\)
							0.515264	+	0.857031i	\(0.327694\pi\)
\(47\)	3.54476	+	1.15176i	0.0754204	+	0.0245056i	0.346484	−	0.938056i	\(-0.387375\pi\)
							−0.271064	+	0.962561i	\(0.587375\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.j.a.11.13	yes	80
3.2	odd	2	inner	150.3.j.a.11.10	✓	80
25.16	even	5	inner	150.3.j.a.41.10	yes	80
75.41	odd	10	inner	150.3.j.a.41.13	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.j.a.11.10	✓	80	3.2	odd	2	inner
150.3.j.a.11.13	yes	80	1.1	even	1	trivial
150.3.j.a.41.10	yes	80	25.16	even	5	inner
150.3.j.a.41.13	yes	80	75.41	odd	10	inner