Embedded newform 1800.4.a.v.1.1

• Label: 1800.4.a.v.1.1
• Level: $1800$
• Weight: $4$
• Character: 1800.1
• Self dual: yes
• Analytic conductor: $106.203$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(106.203438010\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 600)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1800.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+5.00000 q^{7} -14.0000 q^{11} +1.00000 q^{13} -46.0000 q^{17} +19.0000 q^{19} +46.0000 q^{23} -14.0000 q^{29} +133.000 q^{31} +258.000 q^{37} -84.0000 q^{41} -167.000 q^{43} -410.000 q^{47} -318.000 q^{49} -456.000 q^{53} +194.000 q^{59} -17.0000 q^{61} +653.000 q^{67} -828.000 q^{71} +570.000 q^{73} -70.0000 q^{77} -552.000 q^{79} -142.000 q^{83} +1104.00 q^{89} +5.00000 q^{91} +841.000 q^{97} +O(q^{100})\)

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
\(5\)	0	0
			\(7\)	5.00000	0.269975	0.134987	−	0.990847i	\(-0.456901\pi\)
0.134987	+	0.990847i				\(0.456901\pi\)
\(11\)	−14.0000	−0.383742	−0.191871	−	0.981420i	\(-0.561455\pi\)
			−0.191871	+	0.981420i	\(0.561455\pi\)
\(13\)	1.00000	0.0213346	0.0106673	−	0.999943i	\(-0.496604\pi\)
			0.0106673	+	0.999943i	\(0.496604\pi\)
\(17\)	−46.0000	−0.656273	−0.328136	−	0.944630i	\(-0.606421\pi\)
			−0.328136	+	0.944630i	\(0.606421\pi\)
\(19\)	19.0000	0.229416	0.114708	−	0.993399i	\(-0.463407\pi\)
			0.114708	+	0.993399i	\(0.463407\pi\)
\(23\)	46.0000	0.417029	0.208514	−	0.978019i	\(-0.433137\pi\)
			0.208514	+	0.978019i	\(0.433137\pi\)
\(29\)	−14.0000	−0.0896460	−0.0448230	−	0.998995i	\(-0.514272\pi\)
			−0.0448230	+	0.998995i	\(0.514272\pi\)
\(31\)	133.000	0.770565	0.385282	−	0.922799i	\(-0.374104\pi\)
			0.385282	+	0.922799i	\(0.374104\pi\)
\(37\)	258.000	1.14635	0.573175	−	0.819433i	\(-0.305712\pi\)
			0.573175	+	0.819433i	\(0.305712\pi\)
\(41\)	−84.0000	−0.319966	−0.159983	−	0.987120i	\(-0.551144\pi\)
			−0.159983	+	0.987120i	\(0.551144\pi\)
\(43\)	−167.000	−0.592262	−0.296131	−	0.955147i	\(-0.595696\pi\)
			−0.296131	+	0.955147i	\(0.595696\pi\)
\(47\)	−410.000	−1.27244	−0.636220	−	0.771508i	\(-0.719503\pi\)
			−0.636220	+	0.771508i	\(0.719503\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.v.1.1		1
3.2	odd	2		600.4.a.n.1.1	yes	1
5.2	odd	4		1800.4.f.l.649.2		2
5.3	odd	4		1800.4.f.l.649.1		2
5.4	even	2		1800.4.a.m.1.1		1
12.11	even	2		1200.4.a.g.1.1		1
15.2	even	4		600.4.f.e.49.1		2
15.8	even	4		600.4.f.e.49.2		2
15.14	odd	2		600.4.a.e.1.1	✓	1
60.23	odd	4		1200.4.f.i.49.1		2
60.47	odd	4		1200.4.f.i.49.2		2
60.59	even	2		1200.4.a.bd.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.e.1.1	✓	1	15.14	odd	2
600.4.a.n.1.1	yes	1	3.2	odd	2
600.4.f.e.49.1		2	15.2	even	4
600.4.f.e.49.2		2	15.8	even	4
1200.4.a.g.1.1		1	12.11	even	2
1200.4.a.bd.1.1		1	60.59	even	2
1200.4.f.i.49.1		2	60.23	odd	4
1200.4.f.i.49.2		2	60.47	odd	4
1800.4.a.m.1.1		1	5.4	even	2
1800.4.a.v.1.1		1	1.1	even	1	trivial
1800.4.f.l.649.1		2	5.3	odd	4
1800.4.f.l.649.2		2	5.2	odd	4