Embedded newform 150.3.i.a.119.5

• Label: 150.3.i.a.119.5
• Level: $150$
• Weight: $3$
• Character: 150.119
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.i (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			119.5
Character	\(\chi\)	\(=\)	150.119
Dual form			150.3.i.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.437016 - 1.34500i) q^{2} +(-0.0352727 + 2.99979i) q^{3} +(-1.61803 + 1.17557i) q^{4} +(-4.11716 - 2.83707i) q^{5} +(4.05013 - 1.26352i) q^{6} -2.70868i q^{7} +(2.28825 + 1.66251i) q^{8} +(-8.99751 - 0.211621i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.437016	−	1.34500i	−0.218508	−	0.672499i
							\(3\)	−0.0352727	+	2.99979i	−0.0117576	+	0.999931i
\(5\)	−4.11716	−	2.83707i	−0.823432	−	0.567415i
							\(7\)		−	2.70868i		−	0.386955i	−0.981105	−	0.193477i	\(-0.938023\pi\)
0.981105	−	0.193477i	\(-0.0619766\pi\)
\(11\)	−1.51542	+	0.492390i	−0.137765	+	0.0447627i	−0.377088	−	0.926177i	\(-0.623075\pi\)
							0.239323	+	0.970940i	\(0.423075\pi\)
\(13\)	−15.0748	−	4.89810i	−1.15960	−	0.376777i	−0.334848	−	0.942272i	\(-0.608685\pi\)
							−0.824751	+	0.565495i	\(0.808685\pi\)
\(17\)	−18.8231	−	13.6758i	−1.10724	−	0.804460i	−0.125016	−	0.992155i	\(-0.539898\pi\)
							−0.982227	+	0.187695i	\(0.939898\pi\)
\(19\)	−19.4318	−	14.1180i	−1.02273	−	0.743054i	−0.0558866	−	0.998437i	\(-0.517799\pi\)
							−0.966840	+	0.255383i	\(0.917799\pi\)
\(23\)	3.76161	+	11.5770i	0.163548	+	0.503350i	0.998926	−	0.0463259i	\(-0.0147513\pi\)
							−0.835378	+	0.549676i	\(0.814751\pi\)
\(29\)	21.0025	+	28.9075i	0.724225	+	0.996810i	0.999373	+	0.0354080i	\(0.0112731\pi\)
							−0.275148	+	0.961402i	\(0.588727\pi\)
\(31\)	17.8786	+	12.9896i	0.576730	+	0.419019i	0.837544	−	0.546370i	\(-0.183991\pi\)
							−0.260814	+	0.965389i	\(0.583991\pi\)
\(37\)	−13.8928	−	4.51404i	−0.375481	−	0.122001i	0.115196	−	0.993343i	\(-0.463250\pi\)
							−0.490677	+	0.871342i	\(0.663250\pi\)
\(41\)	21.4020	+	6.95393i	0.522000	+	0.169608i	0.558153	−	0.829738i	\(-0.311510\pi\)
							−0.0361530	+	0.999346i	\(0.511510\pi\)
\(43\)		−	50.7907i		−	1.18118i	−0.806972	−	0.590590i	\(-0.798895\pi\)
							0.806972	−	0.590590i	\(-0.201105\pi\)
\(47\)	−37.9194	+	27.5500i	−0.806795	+	0.586171i	−0.912900	−	0.408184i	\(-0.866162\pi\)
							0.106105	+	0.994355i	\(0.466162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.119.5	yes	80
3.2	odd	2	inner	150.3.i.a.119.20	yes	80
25.4	even	10	inner	150.3.i.a.29.20	yes	80
75.29	odd	10	inner	150.3.i.a.29.5	✓	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.5	✓	80	75.29	odd	10	inner
150.3.i.a.29.20	yes	80	25.4	even	10	inner
150.3.i.a.119.5	yes	80	1.1	even	1	trivial
150.3.i.a.119.20	yes	80	3.2	odd	2	inner