Embedded newform 1296.4.a.b.1.1

• Label: 1296.4.a.b.1.1
• Level: $1296$
• Weight: $4$
• Character: 1296.1
• Self dual: yes
• Analytic conductor: $76.466$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(76.4664753674\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1296.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-9.00000 q^{5} +31.0000 q^{7} +15.0000 q^{11} -37.0000 q^{13} -42.0000 q^{17} +28.0000 q^{19} -195.000 q^{23} -44.0000 q^{25} +111.000 q^{29} +205.000 q^{31} -279.000 q^{35} -166.000 q^{37} -261.000 q^{41} +43.0000 q^{43} -177.000 q^{47} +618.000 q^{49} +114.000 q^{53} -135.000 q^{55} -159.000 q^{59} +191.000 q^{61} +333.000 q^{65} +421.000 q^{67} -156.000 q^{71} +182.000 q^{73} +465.000 q^{77} -1133.00 q^{79} +1083.00 q^{83} +378.000 q^{85} -1050.00 q^{89} -1147.00 q^{91} -252.000 q^{95} -901.000 q^{97} +O(q^{100})\)

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
\(5\)	−9.00000	−0.804984				−0.402492	−	0.915423i	\(-0.631856\pi\)
			−0.402492	+	0.915423i	\(0.631856\pi\)
\(7\)	31.0000	1.67384	0.836921	−	0.547323i	\(-0.184353\pi\)
			0.836921	+	0.547323i	\(0.184353\pi\)
\(11\)	15.0000	0.411152	0.205576	−	0.978641i	\(-0.434093\pi\)
			0.205576	+	0.978641i	\(0.434093\pi\)
\(13\)	−37.0000	−0.789381	−0.394691	−	0.918814i	\(-0.629148\pi\)
			−0.394691	+	0.918814i	\(0.629148\pi\)
\(17\)	−42.0000	−0.599206	−0.299603	−	0.954064i	\(-0.596854\pi\)
			−0.299603	+	0.954064i	\(0.596854\pi\)
\(19\)	28.0000	0.338086	0.169043	−	0.985609i	\(-0.445932\pi\)
			0.169043	+	0.985609i	\(0.445932\pi\)
\(23\)	−195.000	−1.76784	−0.883920	−	0.467639i	\(-0.845105\pi\)
			−0.883920	+	0.467639i	\(0.845105\pi\)
\(29\)	111.000	0.710765	0.355382	−	0.934721i	\(-0.384351\pi\)
			0.355382	+	0.934721i	\(0.384351\pi\)
\(31\)	205.000	1.18771	0.593856	−	0.804571i	\(-0.297605\pi\)
			0.593856	+	0.804571i	\(0.297605\pi\)
\(37\)	−166.000	−0.737574	−0.368787	−	0.929514i	\(-0.620227\pi\)
			−0.368787	+	0.929514i	\(0.620227\pi\)
\(41\)	−261.000	−0.994179	−0.497090	−	0.867699i	\(-0.665598\pi\)
			−0.497090	+	0.867699i	\(0.665598\pi\)
\(43\)	43.0000	0.152499	0.0762493	−	0.997089i	\(-0.475706\pi\)
			0.0762493	+	0.997089i	\(0.475706\pi\)
\(47\)	−177.000	−0.549321	−0.274661	−	0.961541i	\(-0.588566\pi\)
			−0.274661	+	0.961541i	\(0.588566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.b.1.1		1
3.2	odd	2		1296.4.a.g.1.1		1
4.3	odd	2		162.4.a.d.1.1		1
9.2	odd	6		432.4.i.a.145.1		2
9.4	even	3		144.4.i.a.97.1		2
9.5	odd	6		432.4.i.a.289.1		2
9.7	even	3		144.4.i.a.49.1		2
12.11	even	2		162.4.a.a.1.1		1
36.7	odd	6		18.4.c.a.13.1	yes	2
36.11	even	6		54.4.c.a.37.1		2
36.23	even	6		54.4.c.a.19.1		2
36.31	odd	6		18.4.c.a.7.1	✓	2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.4.c.a.7.1	✓	2	36.31	odd	6
18.4.c.a.13.1	yes	2	36.7	odd	6
54.4.c.a.19.1		2	36.23	even	6
54.4.c.a.37.1		2	36.11	even	6
144.4.i.a.49.1		2	9.7	even	3
144.4.i.a.97.1		2	9.4	even	3
162.4.a.a.1.1		1	12.11	even	2
162.4.a.d.1.1		1	4.3	odd	2
432.4.i.a.145.1		2	9.2	odd	6
432.4.i.a.289.1		2	9.5	odd	6
1296.4.a.b.1.1		1	1.1	even	1	trivial
1296.4.a.g.1.1		1	3.2	odd	2