Embedded newform 150.4.h.b.19.6

• Label: 150.4.h.b.19.6
• Level: $150$
• Weight: $4$
• Character: 150.19
• Analytic conductor: $8.850$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.85028650086\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			19.6
Character	\(\chi\)	\(=\)	150.19
Dual form			150.4.h.b.79.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.90211 - 0.618034i) q^{2} +(1.76336 + 2.42705i) q^{3} +(3.23607 - 2.35114i) q^{4} +(-6.52419 - 9.07937i) q^{5} +(4.85410 + 3.52671i) q^{6} -35.3218i q^{7} +(4.70228 - 6.47214i) q^{8} +(-2.78115 + 8.55951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 32 q^{4} - 4 q^{5} + 48 q^{6} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.90211	−	0.618034i	0.672499	−	0.218508i
							\(3\)	1.76336	+	2.42705i	0.339358	+	0.467086i
\(5\)	−6.52419	−	9.07937i	−0.583541	−	0.812084i
							\(7\)		−	35.3218i		−	1.90720i	−0.301083	−	0.953598i	\(-0.597348\pi\)
0.301083	−	0.953598i	\(-0.402652\pi\)
\(11\)	8.77278	+	26.9998i	0.240463	+	0.740069i	0.996350	+	0.0853663i	\(0.0272060\pi\)
							−0.755887	+	0.654702i	\(0.772794\pi\)
\(13\)	−74.3723	−	24.1650i	−1.58670	−	0.515551i	−0.622932	−	0.782276i	\(-0.714059\pi\)
							−0.963773	+	0.266725i	\(0.914059\pi\)
\(17\)	67.3184	−	92.6558i	0.960418	−	1.32190i	0.0136761	−	0.999906i	\(-0.495647\pi\)
							0.946741	−	0.321995i	\(-0.104353\pi\)
\(19\)	92.5985	+	67.2767i	1.11808	+	0.812334i	0.983917	−	0.178626i	\(-0.0571651\pi\)
							0.134164	+	0.990959i	\(0.457165\pi\)
\(23\)	67.8058	−	22.0314i	0.614716	−	0.199733i	0.0149234	−	0.999889i	\(-0.495250\pi\)
							0.599793	+	0.800155i	\(0.295250\pi\)
\(29\)	41.9992	−	30.5142i	0.268933	−	0.195391i	−0.445143	−	0.895460i	\(-0.646847\pi\)
							0.714076	+	0.700068i	\(0.246847\pi\)
\(31\)	−8.17260	−	5.93774i	−0.0473497	−	0.0344016i	0.563859	−	0.825871i	\(-0.309316\pi\)
							−0.611208	+	0.791470i	\(0.709316\pi\)
\(37\)	169.370	+	55.0315i	0.752546	+	0.244517i	0.660076	−	0.751199i	\(-0.270524\pi\)
							0.0924696	+	0.995716i	\(0.470524\pi\)
\(41\)	16.7906	−	51.6760i	0.0639572	−	0.196840i	−0.913972	−	0.405778i	\(-0.867001\pi\)
							0.977929	+	0.208938i	\(0.0670007\pi\)
\(43\)			347.424i			1.23213i	0.787695	+	0.616066i	\(0.211274\pi\)
							−0.787695	+	0.616066i	\(0.788726\pi\)
\(47\)	−52.4266	−	72.1590i	−0.162706	−	0.223946i	0.719877	−	0.694101i	\(-0.244198\pi\)
							−0.882584	+	0.470155i	\(0.844198\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.h.b.19.6	✓	32
25.4	even	10	inner	150.4.h.b.79.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.h.b.19.6	✓	32	1.1	even	1	trivial
150.4.h.b.79.6	yes	32	25.4	even	10	inner