Embedded newform 150.3.k.a.67.4

• Label: 150.3.k.a.67.4
• Level: $150$
• Weight: $3$
• Character: 150.67
• 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			67.4
Character	\(\chi\)	\(=\)	150.67
Dual form			150.3.k.a.103.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642040 - 1.26007i) q^{2} +(0.270952 - 1.71073i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(2.55121 - 4.30016i) q^{5} +(-1.98168 - 1.43977i) q^{6} +(-1.41591 - 1.41591i) q^{7} +(-2.79360 + 0.442463i) q^{8} +(-2.85317 - 0.927051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{2} + 8 q^{7} - 16 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.642040	−	1.26007i	0.321020	−	0.630037i
							\(3\)	0.270952	−	1.71073i	0.0903175	−	0.570242i
\(5\)	2.55121	−	4.30016i	0.510241	−	0.860031i
							\(7\)	−1.41591	−	1.41591i	−0.202272	−	0.202272i	0.598701	−	0.800973i	\(-0.295684\pi\)
−0.800973	+	0.598701i	\(0.795684\pi\)
\(11\)	2.20678	+	6.79177i	0.200616	+	0.617434i	0.999865	+	0.0164333i	\(0.00523112\pi\)
							−0.799249	+	0.601001i	\(0.794769\pi\)
\(13\)	−4.04116	−	7.93122i	−0.310858	−	0.610094i	0.681732	−	0.731602i	\(-0.261227\pi\)
							−0.992590	+	0.121508i	\(0.961227\pi\)
\(17\)	−0.278150	−	1.75617i	−0.0163617	−	0.103304i	0.978156	−	0.207872i	\(-0.0666539\pi\)
							−0.994518	+	0.104568i	\(0.966654\pi\)
\(19\)	7.57820	−	10.4305i	0.398853	−	0.548974i	−0.561603	−	0.827407i	\(-0.689815\pi\)
							0.960456	+	0.278433i	\(0.0898151\pi\)
\(23\)	3.87588	+	1.97486i	0.168516	+	0.0858634i	0.536214	−	0.844082i	\(-0.319854\pi\)
							−0.367698	+	0.929945i	\(0.619854\pi\)
\(29\)	3.87575	+	5.33451i	0.133647	+	0.183949i	0.870595	−	0.492000i	\(-0.163734\pi\)
							−0.736949	+	0.675949i	\(0.763734\pi\)
\(31\)	45.6415	+	33.1605i	1.47231	+	1.06969i	0.979937	+	0.199307i	\(0.0638690\pi\)
							0.492369	+	0.870386i	\(0.336131\pi\)
\(37\)	29.1149	−	14.8348i	0.786890	−	0.400940i	−0.0138868	−	0.999904i	\(-0.504420\pi\)
							0.800777	+	0.598963i	\(0.204420\pi\)
\(41\)	−23.1532	+	71.2581i	−0.564711	+	1.73800i	0.104097	+	0.994567i	\(0.466805\pi\)
							−0.668808	+	0.743435i	\(0.733195\pi\)
\(43\)	43.4594	−	43.4594i	1.01068	−	1.01068i	0.0107425	−	0.999942i	\(-0.496580\pi\)
							0.999942	−	0.0107425i	\(-0.00341953\pi\)
\(47\)	57.2627	+	9.06951i	1.21835	+	0.192968i	0.732313	−	0.680968i	\(-0.238441\pi\)
							0.486041	+	0.873936i	\(0.338441\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.a.67.4	✓	32
25.3	odd	20	inner	150.3.k.a.103.4	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.a.67.4	✓	32	1.1	even	1	trivial
150.3.k.a.103.4	yes	32	25.3	odd	20	inner