Embedded newform 150.6.g.c.61.4

• Label: 150.6.g.c.61.4
• 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			61.4
Character	\(\chi\)	\(=\)	150.61
Dual form			150.6.g.c.91.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.23607 - 2.35114i) q^{2} +(2.78115 - 8.55951i) q^{3} +(4.94427 - 15.2169i) q^{4} +(6.45828 - 55.5274i) q^{5} +(-11.1246 - 34.2380i) q^{6} -133.388 q^{7} +(-19.7771 - 60.8676i) q^{8} +(-65.5304 - 47.6106i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} - 54 q^{3} - 96 q^{4} + 80 q^{5} + 216 q^{6} - 454 q^{7} + 384 q^{8} - 486 q^{9}+O(q^{10}) \)

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
\(5\)	6.45828	−	55.5274i	0.115529	−	0.993304i
							\(7\)	−133.388			−1.02890			−0.514450	−	0.857521i	\(-0.672004\pi\)
−0.514450	+	0.857521i	\(0.672004\pi\)
\(11\)	−411.738	+	299.145i	−1.02598	+	0.745418i	−0.967500	−	0.252871i	\(-0.918625\pi\)
							−0.0584797	+	0.998289i	\(0.518625\pi\)
\(13\)	50.9302	+	37.0030i	0.0835829	+	0.0607265i	0.628792	−	0.777574i	\(-0.283550\pi\)
							−0.545209	+	0.838300i	\(0.683550\pi\)
\(17\)	−76.3130	−	234.867i	−0.0640436	−	0.197106i	0.913915	−	0.405906i	\(-0.133044\pi\)
							−0.977958	+	0.208800i	\(0.933044\pi\)
\(19\)	456.592	+	1405.25i	0.290164	+	0.893034i	0.984803	+	0.173675i	\(0.0555644\pi\)
							−0.694638	+	0.719359i	\(0.744436\pi\)
\(23\)	1557.29	−	1131.44i	0.613832	−	0.445975i	−0.236930	−	0.971527i	\(-0.576141\pi\)
							0.850762	+	0.525552i	\(0.176141\pi\)
\(29\)	1193.49	−	3673.20i	0.263527	−	0.811053i	−0.728502	−	0.685044i	\(-0.759783\pi\)
							0.992029	−	0.126009i	\(-0.0402169\pi\)
\(31\)	1353.32	+	4165.10i	0.252928	+	0.778432i	0.994231	+	0.107262i	\(0.0342082\pi\)
							−0.741303	+	0.671171i	\(0.765792\pi\)
\(37\)	−5790.31	−	4206.91i	−0.695341	−	0.505195i	0.183071	−	0.983100i	\(-0.441396\pi\)
							−0.878411	+	0.477905i	\(0.841396\pi\)
\(41\)	−13206.3	−	9594.93i	−1.22693	−	0.891420i	−0.230278	−	0.973125i	\(-0.573963\pi\)
							−0.996657	+	0.0817051i	\(0.973963\pi\)
\(43\)	−6631.14			−0.546912			−0.273456	−	0.961885i	\(-0.588167\pi\)
							−0.273456	+	0.961885i	\(0.588167\pi\)
\(47\)	−2693.42	+	8289.50i	−0.177852	+	0.547373i	−0.999752	−	0.0222588i	\(-0.992914\pi\)
							0.821900	+	0.569632i	\(0.192914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.g.c.61.4	✓	24
25.16	even	5	inner	150.6.g.c.91.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.g.c.61.4	✓	24	1.1	even	1	trivial
150.6.g.c.91.4	yes	24	25.16	even	5	inner