Embedded newform 150.3.i.a.29.5

• Label: 150.3.i.a.29.5
• Level: $150$
• Weight: $3$
• Character: 150.29
• 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			29.5
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.5

$q$-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 \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.5	✓	80	1.1	even	1	trivial
150.3.i.a.29.20	yes	80	3.2	odd	2	inner
150.3.i.a.119.5	yes	80	75.44	odd	10	inner
150.3.i.a.119.20	yes	80	25.19	even	10	inner