Embedded newform 150.3.i.a.29.7

• Label: 150.3.i.a.29.7
• Level: $150$
• Weight: $3$
• Character: 150.29
• 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.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			29.7
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.437016 + 1.34500i) q^{2} +(1.95554 - 2.27505i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(4.48538 + 2.20939i) q^{5} +(2.20534 + 3.62443i) q^{6} +0.464101i q^{7} +(2.28825 - 1.66251i) q^{8} +(-1.35173 - 8.89791i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	1.95554	−	2.27505i	0.651847	−	0.758351i
\(5\)	4.48538	+	2.20939i	0.897076	+	0.441877i
							\(7\)			0.464101i			0.0663001i	0.999450	+	0.0331500i	\(0.0105539\pi\)
−0.999450	+	0.0331500i	\(0.989446\pi\)
\(11\)	6.11171	+	1.98581i	0.555610	+	0.180528i	0.573345	−	0.819314i	\(-0.305646\pi\)
							−0.0177352	+	0.999843i	\(0.505646\pi\)
\(13\)	9.25875	−	3.00835i	0.712212	−	0.231412i	0.0695685	−	0.997577i	\(-0.477838\pi\)
							0.642643	+	0.766166i	\(0.277838\pi\)
\(17\)	0.169413	−	0.123086i	0.00996545	−	0.00724032i	−0.582791	−	0.812622i	\(-0.698040\pi\)
							0.592757	+	0.805381i	\(0.298040\pi\)
\(19\)	12.7593	−	9.27017i	0.671542	−	0.487904i	−0.198999	−	0.980000i	\(-0.563769\pi\)
							0.870541	+	0.492096i	\(0.163769\pi\)
\(23\)	−1.62130	+	4.98985i	−0.0704913	+	0.216950i	−0.980096	−	0.198525i	\(-0.936385\pi\)
							0.909604	+	0.415475i	\(0.136385\pi\)
\(29\)	−27.0306	+	37.2044i	−0.932088	+	1.28291i	0.0269516	+	0.999637i	\(0.491420\pi\)
							−0.959040	+	0.283272i	\(0.908580\pi\)
\(31\)	1.41140	−	1.02544i	0.0455291	−	0.0330789i	−0.564788	−	0.825236i	\(-0.691042\pi\)
							0.610317	+	0.792157i	\(0.291042\pi\)
\(37\)	−41.4482	+	13.4673i	−1.12022	+	0.363982i	−0.809852	−	0.586634i	\(-0.800453\pi\)
							−0.310370	+	0.950616i	\(0.600453\pi\)
\(41\)	−20.5508	+	6.67735i	−0.501238	+	0.162862i	−0.548714	−	0.836010i	\(-0.684882\pi\)
							0.0474756	+	0.998872i	\(0.484882\pi\)
\(43\)		−	61.5090i		−	1.43044i	−0.698899	−	0.715221i	\(-0.746326\pi\)
							0.698899	−	0.715221i	\(-0.253674\pi\)
\(47\)	−45.2661	−	32.8877i	−0.963108	−	0.699739i	−0.00923741	−	0.999957i	\(-0.502940\pi\)
							−0.953870	+	0.300219i	\(0.902940\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.29.7	✓	80
3.2	odd	2	inner	150.3.i.a.29.17	yes	80
25.19	even	10	inner	150.3.i.a.119.17	yes	80
75.44	odd	10	inner	150.3.i.a.119.7	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.7	✓	80	1.1	even	1	trivial
150.3.i.a.29.17	yes	80	3.2	odd	2	inner
150.3.i.a.119.7	yes	80	75.44	odd	10	inner
150.3.i.a.119.17	yes	80	25.19	even	10	inner