Embedded newform 432.6.a.g.1.1

• Label: 432.6.a.g.1.1
• Level: $432$
• Weight: $6$
• Character: 432.1
• Self dual: yes
• Analytic conductor: $69.286$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	432.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(69.2858101592\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	432.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+24.0000 q^{5} -77.0000 q^{7} -408.000 q^{11} +89.0000 q^{13} +2088.00 q^{17} +2617.00 q^{19} -1752.00 q^{23} -2549.00 q^{25} -7296.00 q^{29} -2348.00 q^{31} -1848.00 q^{35} -4993.00 q^{37} -6528.00 q^{41} +6232.00 q^{43} +29832.0 q^{47} -10878.0 q^{49} +22608.0 q^{53} -9792.00 q^{55} -19608.0 q^{59} -22045.0 q^{61} +2136.00 q^{65} -48131.0 q^{67} -51120.0 q^{71} +30737.0 q^{73} +31416.0 q^{77} -38219.0 q^{79} -8112.00 q^{83} +50112.0 q^{85} -44280.0 q^{89} -6853.00 q^{91} +62808.0 q^{95} -136651. q^{97} +O(q^{100})\)

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	0
			\(3\)	0	0
\(5\)	24.0000	0.429325				0.214663	−	0.976688i	\(-0.431135\pi\)
			0.214663	+	0.976688i	\(0.431135\pi\)
\(7\)	−77.0000	−0.593944	−0.296972	−	0.954886i	\(-0.595977\pi\)
			−0.296972	+	0.954886i	\(0.595977\pi\)
\(11\)	−408.000	−1.01667	−0.508333	−	0.861160i	\(-0.669738\pi\)
			−0.508333	+	0.861160i	\(0.669738\pi\)
\(13\)	89.0000	0.146060	0.0730301	−	0.997330i	\(-0.476733\pi\)
			0.0730301	+	0.997330i	\(0.476733\pi\)
\(17\)	2088.00	1.75230	0.876149	−	0.482040i	\(-0.160104\pi\)
			0.876149	+	0.482040i	\(0.160104\pi\)
\(19\)	2617.00	1.66311	0.831553	−	0.555446i	\(-0.187452\pi\)
			0.831553	+	0.555446i	\(0.187452\pi\)
\(23\)	−1752.00	−0.690581	−0.345290	−	0.938496i	\(-0.612220\pi\)
			−0.345290	+	0.938496i	\(0.612220\pi\)
\(29\)	−7296.00	−1.61098	−0.805489	−	0.592610i	\(-0.798097\pi\)
			−0.805489	+	0.592610i	\(0.798097\pi\)
\(31\)	−2348.00	−0.438828	−0.219414	−	0.975632i	\(-0.570414\pi\)
			−0.219414	+	0.975632i	\(0.570414\pi\)
\(37\)	−4993.00	−0.599594	−0.299797	−	0.954003i	\(-0.596919\pi\)
			−0.299797	+	0.954003i	\(0.596919\pi\)
\(41\)	−6528.00	−0.606486	−0.303243	−	0.952913i	\(-0.598069\pi\)
			−0.303243	+	0.952913i	\(0.598069\pi\)
\(43\)	6232.00	0.513992	0.256996	−	0.966412i	\(-0.417267\pi\)
			0.256996	+	0.966412i	\(0.417267\pi\)
\(47\)	29832.0	1.96987	0.984935	−	0.172923i	\(-0.0553212\pi\)
			0.984935	+	0.172923i	\(0.0553212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.a.g.1.1		1
3.2	odd	2		432.6.a.d.1.1		1
4.3	odd	2		54.6.a.e.1.1	yes	1
12.11	even	2		54.6.a.b.1.1	✓	1
36.7	odd	6		162.6.c.c.109.1		2
36.11	even	6		162.6.c.j.109.1		2
36.23	even	6		162.6.c.j.55.1		2
36.31	odd	6		162.6.c.c.55.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.6.a.b.1.1	✓	1	12.11	even	2
54.6.a.e.1.1	yes	1	4.3	odd	2
162.6.c.c.55.1		2	36.31	odd	6
162.6.c.c.109.1		2	36.7	odd	6
162.6.c.j.55.1		2	36.23	even	6
162.6.c.j.109.1		2	36.11	even	6
432.6.a.d.1.1		1	3.2	odd	2
432.6.a.g.1.1		1	1.1	even	1	trivial