Embedded newform 150.3.k.a.13.4

• Label: 150.3.k.a.13.4
• Level: $150$
• Weight: $3$
• Character: 150.13
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	150.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.08720396540\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	150.13
Dual form			150.3.k.a.127.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39680 - 0.221232i) q^{2} +(0.786335 - 1.54327i) q^{3} +(1.90211 - 0.618034i) q^{4} +(4.42349 + 2.33083i) q^{5} +(0.756934 - 2.32960i) q^{6} +(0.284481 - 0.284481i) q^{7} +(2.52015 - 1.28408i) q^{8} +(-1.76336 - 2.42705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{19}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39680	−	0.221232i	0.698401	−	0.110616i
							\(3\)	0.786335	−	1.54327i	0.262112	−	0.514423i
\(5\)	4.42349	+	2.33083i	0.884698	+	0.466165i
							\(7\)	0.284481	−	0.284481i	0.0406401	−	0.0406401i	−0.686495	−	0.727135i	\(-0.740851\pi\)
0.727135	+	0.686495i	\(0.240851\pi\)
\(11\)	0.730582	+	0.530799i	0.0664165	+	0.0482544i	0.620498	−	0.784208i	\(-0.286930\pi\)
							−0.554082	+	0.832462i	\(0.686930\pi\)
\(13\)	−3.27844	−	0.519254i	−0.252188	−	0.0399426i	0.0290606	−	0.999578i	\(-0.490748\pi\)
							−0.281248	+	0.959635i	\(0.590748\pi\)
\(17\)	0.365236	+	0.716816i	0.0214845	+	0.0421656i	0.901499	−	0.432781i	\(-0.142468\pi\)
							−0.880015	+	0.474946i	\(0.842468\pi\)
\(19\)	−6.47160	−	2.10275i	−0.340610	−	0.110671i	0.133717	−	0.991020i	\(-0.457309\pi\)
							−0.474327	+	0.880349i	\(0.657309\pi\)
\(23\)	−3.75220	−	23.6905i	−0.163139	−	1.03002i	−0.924358	−	0.381526i	\(-0.875399\pi\)
							0.761219	−	0.648495i	\(-0.224601\pi\)
\(29\)	−25.0003	+	8.12309i	−0.862080	+	0.280107i	−0.706497	−	0.707716i	\(-0.749726\pi\)
							−0.155583	+	0.987823i	\(0.549726\pi\)
\(31\)	−6.76707	+	20.8269i	−0.218293	+	0.671836i	0.780611	+	0.625017i	\(0.214908\pi\)
							−0.998903	+	0.0468181i	\(0.985092\pi\)
\(37\)	−10.7495	+	67.8699i	−0.290528	+	1.83432i	0.221266	+	0.975213i	\(0.428981\pi\)
							−0.511794	+	0.859108i	\(0.671019\pi\)
\(41\)	−32.7794	+	23.8157i	−0.799498	+	0.580870i	−0.910767	−	0.412921i	\(-0.864509\pi\)
							0.111269	+	0.993790i	\(0.464509\pi\)
\(43\)	−23.3195	−	23.3195i	−0.542314	−	0.542314i	0.381893	−	0.924207i	\(-0.375272\pi\)
							−0.924207	+	0.381893i	\(0.875272\pi\)
\(47\)	−8.44318	−	4.30202i	−0.179642	−	0.0915322i	0.361857	−	0.932234i	\(-0.382143\pi\)
							−0.541499	+	0.840701i	\(0.682143\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.a.13.4	✓	32
25.2	odd	20	inner	150.3.k.a.127.4	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.a.13.4	✓	32	1.1	even	1	trivial
150.3.k.a.127.4	yes	32	25.2	odd	20	inner