Embedded newform 150.3.k.b.67.6

• Label: 150.3.k.b.67.6
• Level: $150$
• Weight: $3$
• Character: 150.67
• Analytic conductor: $4.087$
• Analytic rank: $0$
• Dimension: $48$
• 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:	\(48\)
Relative dimension:	\(6\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			67.6
Character	\(\chi\)	\(=\)	150.67
Dual form			150.3.k.b.103.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642040 + 1.26007i) q^{2} +(0.270952 - 1.71073i) q^{3} +(-1.17557 - 1.61803i) q^{4} +(4.98067 - 0.439226i) q^{5} +(1.98168 + 1.43977i) q^{6} +(-7.02992 - 7.02992i) q^{7} +(2.79360 - 0.442463i) q^{8} +(-2.85317 - 0.927051i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 12 q^{2} + 8 q^{7} + 24 q^{8}+O(q^{10}) \)

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
\(5\)	4.98067	−	0.439226i	0.996134	−	0.0878452i
							\(7\)	−7.02992	−	7.02992i	−1.00427	−	1.00427i	−0.999991	−	0.00428345i	\(-0.998637\pi\)
−0.00428345	−	0.999991i	\(-0.501363\pi\)
\(11\)	−5.28977	−	16.2802i	−0.480888	−	1.48002i	−0.837849	−	0.545902i	\(-0.816187\pi\)
							0.356961	−	0.934119i	\(-0.383813\pi\)
\(13\)	7.35327	+	14.4316i	0.565636	+	1.11012i	0.979811	+	0.199926i	\(0.0640702\pi\)
							−0.414175	+	0.910197i	\(0.635930\pi\)
\(17\)	−2.52100	−	15.9170i	−0.148294	−	0.936291i	−0.943841	−	0.330400i	\(-0.892816\pi\)
							0.795547	−	0.605892i	\(-0.207184\pi\)
\(19\)	17.1531	−	23.6092i	0.902793	−	1.24259i	−0.0667761	−	0.997768i	\(-0.521271\pi\)
							0.969569	−	0.244819i	\(-0.0787287\pi\)
\(23\)	0.511194	+	0.260466i	0.0222258	+	0.0113246i	0.465068	−	0.885275i	\(-0.346030\pi\)
							−0.442842	+	0.896600i	\(0.646030\pi\)
\(29\)	8.10712	+	11.1585i	0.279556	+	0.384776i	0.925587	−	0.378536i	\(-0.123572\pi\)
							−0.646031	+	0.763311i	\(0.723572\pi\)
\(31\)	35.5624	+	25.8376i	1.14717	+	0.833471i	0.988103	−	0.153797i	\(-0.0491500\pi\)
							0.159071	+	0.987267i	\(0.449150\pi\)
\(37\)	−51.1744	+	26.0746i	−1.38309	+	0.704720i	−0.977814	−	0.209474i	\(-0.932825\pi\)
							−0.405277	+	0.914194i	\(0.632825\pi\)
\(41\)	10.8365	−	33.3512i	0.264304	−	0.813444i	−0.727549	−	0.686056i	\(-0.759341\pi\)
							0.991853	−	0.127388i	\(-0.0406593\pi\)
\(43\)	−51.5445	+	51.5445i	−1.19871	+	1.19871i	−0.224156	+	0.974553i	\(0.571963\pi\)
							−0.974553	+	0.224156i	\(0.928037\pi\)
\(47\)	−2.96845	−	0.470156i	−0.0631585	−	0.0100033i	0.124775	−	0.992185i	\(-0.460179\pi\)
							−0.187934	+	0.982182i	\(0.560179\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.3.k.b.67.6	✓	48
25.3	odd	20	inner	150.3.k.b.103.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.3.k.b.67.6	✓	48	1.1	even	1	trivial
150.3.k.b.103.6	yes	48	25.3	odd	20	inner