Embedded newform 468.4.a.a.1.1

• Label: 468.4.a.a.1.1
• Level: $468$
• Weight: $4$
• Character: 468.1
• Self dual: yes
• Analytic conductor: $27.613$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	468.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(27.6128938827\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	468.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{5} -32.0000 q^{7} +68.0000 q^{11} +13.0000 q^{13} +14.0000 q^{17} +4.00000 q^{19} -72.0000 q^{23} -121.000 q^{25} -102.000 q^{29} -136.000 q^{31} -64.0000 q^{35} -386.000 q^{37} -250.000 q^{41} -140.000 q^{43} +296.000 q^{47} +681.000 q^{49} -526.000 q^{53} +136.000 q^{55} -332.000 q^{59} -410.000 q^{61} +26.0000 q^{65} +596.000 q^{67} +880.000 q^{71} +506.000 q^{73} -2176.00 q^{77} -640.000 q^{79} -1380.00 q^{83} +28.0000 q^{85} -1450.00 q^{89} -416.000 q^{91} +8.00000 q^{95} -446.000 q^{97} +O(q^{100})\)

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
\(5\)	2.00000	0.178885				0.0894427	−	0.995992i	\(-0.471491\pi\)
			0.0894427	+	0.995992i	\(0.471491\pi\)
\(7\)	−32.0000	−1.72784	−0.863919	−	0.503631i	\(-0.831997\pi\)
			−0.863919	+	0.503631i	\(0.831997\pi\)
\(11\)	68.0000	1.86389	0.931944	−	0.362602i	\(-0.118111\pi\)
			0.931944	+	0.362602i	\(0.118111\pi\)
\(13\)	13.0000	0.277350
			\(17\)	14.0000	0.199735	0.0998676	−	0.995001i	\(-0.468158\pi\)
0.0998676	+	0.995001i				\(0.468158\pi\)
\(19\)	4.00000	0.0482980	0.0241490	−	0.999708i	\(-0.492312\pi\)
			0.0241490	+	0.999708i	\(0.492312\pi\)
\(23\)	−72.0000	−0.652741	−0.326370	−	0.945242i	\(-0.605826\pi\)
			−0.326370	+	0.945242i	\(0.605826\pi\)
\(29\)	−102.000	−0.653135	−0.326568	−	0.945174i	\(-0.605892\pi\)
			−0.326568	+	0.945174i	\(0.605892\pi\)
\(31\)	−136.000	−0.787946	−0.393973	−	0.919122i	\(-0.628900\pi\)
			−0.393973	+	0.919122i	\(0.628900\pi\)
\(37\)	−386.000	−1.71508	−0.857541	−	0.514416i	\(-0.828009\pi\)
			−0.857541	+	0.514416i	\(0.828009\pi\)
\(41\)	−250.000	−0.952279	−0.476140	−	0.879370i	\(-0.657964\pi\)
			−0.476140	+	0.879370i	\(0.657964\pi\)
\(43\)	−140.000	−0.496507	−0.248253	−	0.968695i	\(-0.579857\pi\)
			−0.248253	+	0.968695i	\(0.579857\pi\)
\(47\)	296.000	0.918639	0.459320	−	0.888271i	\(-0.348093\pi\)
			0.459320	+	0.888271i	\(0.348093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.4.a.a.1.1		1
3.2	odd	2		156.4.a.b.1.1	✓	1
4.3	odd	2		1872.4.a.i.1.1		1
12.11	even	2		624.4.a.b.1.1		1
24.5	odd	2		2496.4.a.d.1.1		1
24.11	even	2		2496.4.a.m.1.1		1
39.5	even	4		2028.4.b.d.337.2		2
39.8	even	4		2028.4.b.d.337.1		2
39.38	odd	2		2028.4.a.b.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.b.1.1	✓	1	3.2	odd	2
468.4.a.a.1.1		1	1.1	even	1	trivial
624.4.a.b.1.1		1	12.11	even	2
1872.4.a.i.1.1		1	4.3	odd	2
2028.4.a.b.1.1		1	39.38	odd	2
2028.4.b.d.337.1		2	39.8	even	4
2028.4.b.d.337.2		2	39.5	even	4
2496.4.a.d.1.1		1	24.5	odd	2
2496.4.a.m.1.1		1	24.11	even	2