Embedded newform 150.3.i.a.29.6

• Label: 150.3.i.a.29.6
• 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			29.6
Character	\(\chi\)	\(=\)	150.29
Dual form			150.3.i.a.119.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.437016 + 1.34500i) q^{2} +(-0.0189915 + 2.99994i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-4.54494 - 2.08411i) q^{5} +(-4.02661 - 1.33657i) q^{6} +1.94466i q^{7} +(2.28825 - 1.66251i) q^{8} +(-8.99928 - 0.113946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 40 q^{4} + 20 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.0189915	+	2.99994i	−0.00633049	+	0.999980i
\(5\)	−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	−0.473512	−	0.880788i	\(-0.657014\pi\)
							−0.900793	+	0.434249i	\(0.857014\pi\)
\(13\)	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	−0.902328	−	0.431051i	\(-0.858143\pi\)
							0.131119	+	0.991367i	\(0.458143\pi\)
\(19\)	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	\(-0.752514\pi\)
							0.988893	−	0.148630i	\(-0.0474863\pi\)
\(29\)	−21.0884	+	29.0257i	−0.727187	+	1.00089i	0.272068	+	0.962278i	\(0.412292\pi\)
							−0.999254	+	0.0386084i	\(0.987708\pi\)
\(31\)	−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	−0.651032	−	0.759050i	\(-0.725664\pi\)
							−0.0805380	+	0.996752i	\(0.525664\pi\)
\(43\)			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	0.709244	−	0.704963i	\(-0.249036\pi\)
							−0.451291	+	0.892377i	\(0.649036\pi\)

(See \(a_n\) instead)

Twists

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

    

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