Embedded newform 1656.4.a.c.1.1

• Label: 1656.4.a.c.1.1
• Level: $1656$
• Weight: $4$
• Character: 1656.1
• Self dual: yes
• Analytic conductor: $97.707$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1656 = 2^{3} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1656.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(97.7071629695\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1656.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{5} -4.00000 q^{7} -26.0000 q^{11} +70.0000 q^{13} -94.0000 q^{17} +54.0000 q^{19} +23.0000 q^{23} -109.000 q^{25} +86.0000 q^{29} -144.000 q^{31} -16.0000 q^{35} -172.000 q^{37} +42.0000 q^{41} +386.000 q^{43} +80.0000 q^{47} -327.000 q^{49} +108.000 q^{53} -104.000 q^{55} -164.000 q^{59} -400.000 q^{61} +280.000 q^{65} +398.000 q^{67} +320.000 q^{71} -810.000 q^{73} +104.000 q^{77} -204.000 q^{79} -102.000 q^{83} -376.000 q^{85} -1018.00 q^{89} -280.000 q^{91} +216.000 q^{95} -1370.00 q^{97} +O(q^{100})\)

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
\(5\)	4.00000	0.357771				0.178885	−	0.983870i	\(-0.442751\pi\)
			0.178885	+	0.983870i	\(0.442751\pi\)
\(7\)	−4.00000	−0.215980	−0.107990	−	0.994152i	\(-0.534441\pi\)
			−0.107990	+	0.994152i	\(0.534441\pi\)
\(11\)	−26.0000	−0.712663	−0.356332	−	0.934360i	\(-0.615973\pi\)
			−0.356332	+	0.934360i	\(0.615973\pi\)
\(13\)	70.0000	1.49342	0.746712	−	0.665148i	\(-0.231631\pi\)
			0.746712	+	0.665148i	\(0.231631\pi\)
\(17\)	−94.0000	−1.34108	−0.670540	−	0.741874i	\(-0.733937\pi\)
			−0.670540	+	0.741874i	\(0.733937\pi\)
\(19\)	54.0000	0.652024	0.326012	−	0.945366i	\(-0.394295\pi\)
			0.326012	+	0.945366i	\(0.394295\pi\)
\(23\)	23.0000	0.208514
			\(29\)	86.0000	0.550683	0.275341	−	0.961347i	\(-0.411209\pi\)
0.275341	+	0.961347i				\(0.411209\pi\)
\(31\)	−144.000	−0.834296	−0.417148	−	0.908839i	\(-0.636970\pi\)
			−0.417148	+	0.908839i	\(0.636970\pi\)
\(37\)	−172.000	−0.764233	−0.382117	−	0.924114i	\(-0.624805\pi\)
			−0.382117	+	0.924114i	\(0.624805\pi\)
\(41\)	42.0000	0.159983	0.0799914	−	0.996796i	\(-0.474511\pi\)
			0.0799914	+	0.996796i	\(0.474511\pi\)
\(43\)	386.000	1.36894	0.684470	−	0.729041i	\(-0.260034\pi\)
			0.684470	+	0.729041i	\(0.260034\pi\)
\(47\)	80.0000	0.248281	0.124140	−	0.992265i	\(-0.460383\pi\)
			0.124140	+	0.992265i	\(0.460383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1656.4.a.c.1.1		1
3.2	odd	2		184.4.a.b.1.1	✓	1
12.11	even	2		368.4.a.a.1.1		1
24.5	odd	2		1472.4.a.b.1.1		1
24.11	even	2		1472.4.a.i.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.b.1.1	✓	1	3.2	odd	2
368.4.a.a.1.1		1	12.11	even	2
1472.4.a.b.1.1		1	24.5	odd	2
1472.4.a.i.1.1		1	24.11	even	2
1656.4.a.c.1.1		1	1.1	even	1	trivial