Embedded newform 1764.4.a.i.1.1

• Label: 1764.4.a.i.1.1
• Level: $1764$
• Weight: $4$
• Character: 1764.1
• Self dual: yes
• Analytic conductor: $104.079$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(104.079369250\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 588)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1764.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+4.00000 q^{5} +20.0000 q^{11} +4.00000 q^{13} +24.0000 q^{17} -44.0000 q^{19} -72.0000 q^{23} -109.000 q^{25} +38.0000 q^{29} -184.000 q^{31} -30.0000 q^{37} -216.000 q^{41} -164.000 q^{43} +520.000 q^{47} +146.000 q^{53} +80.0000 q^{55} +460.000 q^{59} -628.000 q^{61} +16.0000 q^{65} +556.000 q^{67} -592.000 q^{71} -1024.00 q^{73} -104.000 q^{79} -324.000 q^{83} +96.0000 q^{85} +896.000 q^{89} -176.000 q^{95} +920.000 q^{97} +O(q^{100})\)

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
\(5\)	4.00000	0.357771				0.178885	−	0.983870i	\(-0.442751\pi\)
			0.178885	+	0.983870i	\(0.442751\pi\)
\(7\)	0	0
			\(11\)	20.0000	0.548202	0.274101	−	0.961701i	\(-0.411620\pi\)
0.274101	+	0.961701i				\(0.411620\pi\)
\(13\)	4.00000	0.0853385	0.0426692	−	0.999089i	\(-0.486414\pi\)
			0.0426692	+	0.999089i	\(0.486414\pi\)
\(17\)	24.0000	0.342403	0.171202	−	0.985236i	\(-0.445235\pi\)
			0.171202	+	0.985236i	\(0.445235\pi\)
\(19\)	−44.0000	−0.531279	−0.265639	−	0.964072i	\(-0.585583\pi\)
			−0.265639	+	0.964072i	\(0.585583\pi\)
\(23\)	−72.0000	−0.652741	−0.326370	−	0.945242i	\(-0.605826\pi\)
			−0.326370	+	0.945242i	\(0.605826\pi\)
\(29\)	38.0000	0.243325	0.121662	−	0.992572i	\(-0.461177\pi\)
			0.121662	+	0.992572i	\(0.461177\pi\)
\(31\)	−184.000	−1.06604	−0.533022	−	0.846101i	\(-0.678944\pi\)
			−0.533022	+	0.846101i	\(0.678944\pi\)
\(37\)	−30.0000	−0.133296	−0.0666482	−	0.997777i	\(-0.521231\pi\)
			−0.0666482	+	0.997777i	\(0.521231\pi\)
\(41\)	−216.000	−0.822769	−0.411385	−	0.911462i	\(-0.634955\pi\)
			−0.411385	+	0.911462i	\(0.634955\pi\)
\(43\)	−164.000	−0.581622	−0.290811	−	0.956780i	\(-0.593925\pi\)
			−0.290811	+	0.956780i	\(0.593925\pi\)
\(47\)	520.000	1.61383	0.806913	−	0.590671i	\(-0.201137\pi\)
			0.806913	+	0.590671i	\(0.201137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.a.i.1.1		1
3.2	odd	2		588.4.a.e.1.1	yes	1
7.2	even	3		1764.4.k.g.361.1		2
7.3	odd	6		1764.4.k.j.1549.1		2
7.4	even	3		1764.4.k.g.1549.1		2
7.5	odd	6		1764.4.k.j.361.1		2
7.6	odd	2		1764.4.a.d.1.1		1
12.11	even	2		2352.4.a.g.1.1		1
21.2	odd	6		588.4.i.b.361.1		2
21.5	even	6		588.4.i.g.361.1		2
21.11	odd	6		588.4.i.b.373.1		2
21.17	even	6		588.4.i.g.373.1		2
21.20	even	2		588.4.a.b.1.1	✓	1
84.83	odd	2		2352.4.a.be.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.4.a.b.1.1	✓	1	21.20	even	2
588.4.a.e.1.1	yes	1	3.2	odd	2
588.4.i.b.361.1		2	21.2	odd	6
588.4.i.b.373.1		2	21.11	odd	6
588.4.i.g.361.1		2	21.5	even	6
588.4.i.g.373.1		2	21.17	even	6
1764.4.a.d.1.1		1	7.6	odd	2
1764.4.a.i.1.1		1	1.1	even	1	trivial
1764.4.k.g.361.1		2	7.2	even	3
1764.4.k.g.1549.1		2	7.4	even	3
1764.4.k.j.361.1		2	7.5	odd	6
1764.4.k.j.1549.1		2	7.3	odd	6
2352.4.a.g.1.1		1	12.11	even	2
2352.4.a.be.1.1		1	84.83	odd	2