Embedded newform 150.3.i.a.29.11

• Label: 150.3.i.a.29.11
• 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			29.11
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.437016 - 1.34500i) q^{2} +(-2.84724 + 0.945094i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(4.54494 + 2.08411i) q^{5} +(0.0268580 + 4.24256i) q^{6} +1.94466i q^{7} +(-2.28825 + 1.66251i) q^{8} +(7.21359 - 5.38183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−2.84724	+	0.945094i	−0.949081	+	0.315031i
\(5\)	4.54494	+	2.08411i	0.908988	+	0.416823i
							\(7\)			1.94466i			0.277809i	0.990306	+	0.138904i	\(0.0443580\pi\)
−0.990306	+	0.138904i	\(0.955642\pi\)
\(11\)	15.1173	+	4.91192i	1.37430	+	0.446539i	0.900793	−	0.434249i	\(-0.142986\pi\)
							0.473512	+	0.880788i	\(0.342986\pi\)
\(13\)	3.25776	−	1.05851i	0.250597	−	0.0814238i	−0.181025	−	0.983478i	\(-0.557942\pi\)
							0.431622	+	0.902055i	\(0.357942\pi\)
\(17\)	13.1105	−	9.52537i	0.771208	−	0.560316i	−0.131119	−	0.991367i	\(-0.541857\pi\)
							0.902328	+	0.431051i	\(0.141857\pi\)
\(19\)	13.0007	−	9.44558i	0.684249	−	0.497136i	−0.190516	−	0.981684i	\(-0.561016\pi\)
							0.874764	+	0.484548i	\(0.161016\pi\)
\(23\)	−6.35316	+	19.5530i	−0.276224	+	0.850131i	0.712669	+	0.701501i	\(0.247486\pi\)
							−0.988893	+	0.148630i	\(0.952514\pi\)
\(29\)	21.0884	−	29.0257i	0.727187	−	1.00089i	−0.272068	−	0.962278i	\(-0.587708\pi\)
							0.999254	−	0.0386084i	\(-0.0122925\pi\)
\(31\)	−47.0929	+	34.2150i	−1.51913	+	1.10371i	−0.557210	+	0.830372i	\(0.688128\pi\)
							−0.961918	+	0.273339i	\(0.911872\pi\)
\(37\)	−44.3666	+	14.4156i	−1.19910	+	0.389610i	−0.839429	−	0.543470i	\(-0.817110\pi\)
							−0.359669	+	0.933080i	\(0.617110\pi\)
\(41\)	29.9944	−	9.74577i	0.731570	−	0.237702i	0.0805380	−	0.996752i	\(-0.474336\pi\)
							0.651032	+	0.759050i	\(0.274336\pi\)
\(43\)			27.2603i			0.633960i	0.948432	+	0.316980i	\(0.102669\pi\)
							−0.948432	+	0.316980i	\(0.897331\pi\)
\(47\)	−12.1238	−	8.80844i	−0.257953	−	0.187414i	0.451291	−	0.892377i	\(-0.350964\pi\)
							−0.709244	+	0.704963i	\(0.750964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.i.a.29.11	yes	80
3.2	odd	2	inner	150.3.i.a.29.6	✓	80
25.19	even	10	inner	150.3.i.a.119.6	yes	80
75.44	odd	10	inner	150.3.i.a.119.11	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.i.a.29.6	✓	80	3.2	odd	2	inner
150.3.i.a.29.11	yes	80	1.1	even	1	trivial
150.3.i.a.119.6	yes	80	25.19	even	10	inner
150.3.i.a.119.11	yes	80	75.44	odd	10	inner