Embedded newform 150.4.g.c.91.2

• Label: 150.4.g.c.91.2
• Level: $150$
• Weight: $4$
• Character: 150.91
• Analytic conductor: $8.850$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.85028650086\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial:	x^16 + 289*x^14 + 30961*x^12 + 1537059*x^10 + 36752711*x^8 + 389532130*x^6 + 1345735885*x^4 + 1301615700*x^2 + 282912400
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4}\cdot 5^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			91.2
Root			\(-4.55041i\) of defining polynomial
Character	\(\chi\)	\(=\)	150.91
Dual form			150.4.g.c.61.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61803 + 1.17557i) q^{2} +(0.927051 + 2.85317i) q^{3} +(1.23607 + 3.80423i) q^{4} +(-2.82602 - 10.8173i) q^{5} +(-1.85410 + 5.70634i) q^{6} +27.7579 q^{7} +(-2.47214 + 7.60845i) q^{8} +(-7.28115 + 5.29007i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{2} - 12 q^{3} - 16 q^{4} + 5 q^{5} + 24 q^{6} + 12 q^{7} + 32 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61803	+	1.17557i	0.572061	+	0.415627i
							\(3\)	0.927051	+	2.85317i	0.178411	+	0.549093i
\(5\)	−2.82602	−	10.8173i	−0.252767	−	0.967527i
							\(7\)	27.7579			1.49879			0.749394	−	0.662124i	\(-0.230345\pi\)
0.749394	+	0.662124i	\(0.230345\pi\)
\(11\)	47.2041	+	34.2958i	1.29387	+	0.940052i	0.999876	−	0.0157554i	\(-0.00501531\pi\)
							0.293994	+	0.955807i	\(0.405015\pi\)
\(13\)	1.79529	−	1.30436i	0.0383019	−	0.0278280i	−0.568470	−	0.822704i	\(-0.692464\pi\)
							0.606772	+	0.794876i	\(0.292464\pi\)
\(17\)	−36.8558	+	113.430i	−0.525814	+	1.61829i	0.236886	+	0.971537i	\(0.423873\pi\)
							−0.762701	+	0.646752i	\(0.776127\pi\)
\(19\)	23.2014	−	71.4065i	0.280145	−	0.862199i	−0.707667	−	0.706547i	\(-0.750252\pi\)
							0.987812	−	0.155652i	\(-0.0497479\pi\)
\(23\)	−40.0128	−	29.0710i	−0.362749	−	0.263553i	0.391449	−	0.920200i	\(-0.371974\pi\)
							−0.754198	+	0.656647i	\(0.771974\pi\)
\(29\)	−23.1082	−	71.1199i	−0.147969	−	0.455401i	0.849412	−	0.527730i	\(-0.176957\pi\)
							−0.997381	+	0.0723292i	\(0.976957\pi\)
\(31\)	23.4473	−	72.1633i	0.135847	−	0.418094i	−0.859874	−	0.510507i	\(-0.829458\pi\)
							0.995721	+	0.0924126i	\(0.0294579\pi\)
\(37\)	195.337	−	141.921i	0.867926	−	0.630585i	−0.0621036	−	0.998070i	\(-0.519781\pi\)
							0.930030	+	0.367485i	\(0.119781\pi\)
\(41\)	37.4725	−	27.2254i	0.142737	−	0.103705i	−0.514125	−	0.857715i	\(-0.671883\pi\)
							0.656862	+	0.754011i	\(0.271883\pi\)
\(43\)	295.540			1.04813			0.524063	−	0.851680i	\(-0.324416\pi\)
							0.524063	+	0.851680i	\(0.324416\pi\)
\(47\)	31.1064	+	95.7357i	0.0965390	+	0.297117i	0.987652	−	0.156665i	\(-0.0500744\pi\)
							−0.891113	+	0.453782i	\(0.850074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.g.c.91.2	yes	16
25.11	even	5	inner	150.4.g.c.61.2	✓	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.g.c.61.2	✓	16	25.11	even	5	inner
150.4.g.c.91.2	yes	16	1.1	even	1	trivial