Embedded newform 150.6.g.c.61.6

• Label: 150.6.g.c.61.6
• Level: $150$
• Weight: $6$
• Character: 150.61
• Analytic conductor: $24.058$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(24.0575729719\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(6\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			61.6
Character	\(\chi\)	\(=\)	150.61
Dual form			150.6.g.c.91.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23607 - 2.35114i) q^{2} +(2.78115 - 8.55951i) q^{3} +(4.94427 - 15.2169i) q^{4} +(54.0113 + 14.4147i) q^{5} +(-11.1246 - 34.2380i) q^{6} +182.018 q^{7} +(-19.7771 - 60.8676i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} - 54 q^{3} - 96 q^{4} + 80 q^{5} + 216 q^{6} - 454 q^{7} + 384 q^{8} - 486 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.23607	−	2.35114i	0.572061	−	0.415627i
							\(3\)	2.78115	−	8.55951i	0.178411	−	0.549093i
\(5\)	54.0113	+	14.4147i	0.966183	+	0.257858i
							\(7\)	182.018			1.40401			0.702003	−	0.712174i	\(-0.252289\pi\)
0.702003	+	0.712174i	\(0.252289\pi\)
\(11\)	0.112375	−	0.0816450i	0.000280019	−	0.000203445i	−0.587645	−	0.809119i	\(-0.699945\pi\)
							0.587925	+	0.808915i	\(0.299945\pi\)
\(13\)	476.041	+	345.864i	0.781242	+	0.567606i	0.905352	−	0.424663i	\(-0.139607\pi\)
							−0.124109	+	0.992269i	\(0.539607\pi\)
\(17\)	214.821	+	661.151i	0.180283	+	0.554853i	0.999835	−	0.0181508i	\(-0.00577791\pi\)
							−0.819553	+	0.573004i	\(0.805778\pi\)
\(19\)	436.563	+	1343.60i	0.277436	+	0.853861i	0.988565	+	0.150799i	\(0.0481845\pi\)
							−0.711128	+	0.703062i	\(0.751816\pi\)
\(23\)	−584.476	+	424.647i	−0.230381	+	0.167382i	−0.696987	−	0.717083i	\(-0.745477\pi\)
							0.466606	+	0.884465i	\(0.345477\pi\)
\(29\)	1772.51	−	5455.21i	0.391375	−	1.20453i	−0.540374	−	0.841425i	\(-0.681717\pi\)
							0.931749	−	0.363103i	\(-0.118283\pi\)
\(31\)	−2230.49	−	6864.74i	−0.416866	−	1.28298i	−0.910571	−	0.413352i	\(-0.864358\pi\)
							0.493706	−	0.869629i	\(-0.335642\pi\)
\(37\)	−11053.4	−	8030.74i	−1.32736	−	0.964386i	−0.999809	−	0.0195544i	\(-0.993775\pi\)
							−0.327555	−	0.944832i	\(-0.606225\pi\)
\(41\)	−8253.29	−	5996.37i	−0.766774	−	0.557094i	0.134206	−	0.990953i	\(-0.457151\pi\)
							−0.900981	+	0.433859i	\(0.857151\pi\)
\(43\)	−112.578			−0.00928501			−0.00464250	−	0.999989i	\(-0.501478\pi\)
							−0.00464250	+	0.999989i	\(0.501478\pi\)
\(47\)	−1790.85	+	5511.68i	−0.118254	+	0.363948i	−0.992612	−	0.121334i	\(-0.961283\pi\)
							0.874358	+	0.485281i	\(0.161283\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.c.61.6	✓	24
25.16	even	5	inner	150.6.g.c.91.6	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.c.61.6	✓	24	1.1	even	1	trivial
150.6.g.c.91.6	yes	24	25.16	even	5	inner