Embedded newform 900.4.a.d.1.1

• Label: 900.4.a.d.1.1
• Level: $900$
• Weight: $4$
• Character: 900.1
• Self dual: yes
• Analytic conductor: $53.102$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(53.1017190052\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 60)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	900.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-22.0000 q^{7} +14.0000 q^{11} +30.0000 q^{13} +62.0000 q^{17} -120.000 q^{19} +188.000 q^{23} -96.0000 q^{29} +184.000 q^{31} -406.000 q^{37} -130.000 q^{41} -148.000 q^{43} +448.000 q^{47} +141.000 q^{49} -414.000 q^{53} -266.000 q^{59} -838.000 q^{61} -248.000 q^{67} -1020.00 q^{71} -484.000 q^{73} -308.000 q^{77} -48.0000 q^{79} +548.000 q^{83} +650.000 q^{89} -660.000 q^{91} +1816.00 q^{97} +O(q^{100})\)

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
\(5\)	0	0
			\(7\)	−22.0000	−1.18789	−0.593944	−	0.804506i	\(-0.702430\pi\)
−0.593944	+	0.804506i				\(0.702430\pi\)
\(11\)	14.0000	0.383742	0.191871	−	0.981420i	\(-0.438545\pi\)
			0.191871	+	0.981420i	\(0.438545\pi\)
\(13\)	30.0000	0.640039	0.320019	−	0.947411i	\(-0.396311\pi\)
			0.320019	+	0.947411i	\(0.396311\pi\)
\(17\)	62.0000	0.884542	0.442271	−	0.896882i	\(-0.354173\pi\)
			0.442271	+	0.896882i	\(0.354173\pi\)
\(19\)	−120.000	−1.44894	−0.724471	−	0.689306i	\(-0.757916\pi\)
			−0.724471	+	0.689306i	\(0.757916\pi\)
\(23\)	188.000	1.70438	0.852189	−	0.523234i	\(-0.175274\pi\)
			0.852189	+	0.523234i	\(0.175274\pi\)
\(29\)	−96.0000	−0.614716	−0.307358	−	0.951594i	\(-0.599445\pi\)
			−0.307358	+	0.951594i	\(0.599445\pi\)
\(31\)	184.000	1.06604	0.533022	−	0.846101i	\(-0.321056\pi\)
			0.533022	+	0.846101i	\(0.321056\pi\)
\(37\)	−406.000	−1.80395	−0.901973	−	0.431793i	\(-0.857881\pi\)
			−0.901973	+	0.431793i	\(0.857881\pi\)
\(41\)	−130.000	−0.495185	−0.247593	−	0.968864i	\(-0.579639\pi\)
			−0.247593	+	0.968864i	\(0.579639\pi\)
\(43\)	−148.000	−0.524879	−0.262439	−	0.964948i	\(-0.584527\pi\)
			−0.262439	+	0.964948i	\(0.584527\pi\)
\(47\)	448.000	1.39037	0.695186	−	0.718830i	\(-0.255322\pi\)
			0.695186	+	0.718830i	\(0.255322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.a.d.1.1		1
3.2	odd	2		300.4.a.f.1.1		1
5.2	odd	4		180.4.d.b.109.2		2
5.3	odd	4		180.4.d.b.109.1		2
5.4	even	2		900.4.a.o.1.1		1
12.11	even	2		1200.4.a.q.1.1		1
15.2	even	4		60.4.d.a.49.1	✓	2
15.8	even	4		60.4.d.a.49.2	yes	2
15.14	odd	2		300.4.a.d.1.1		1
20.3	even	4		720.4.f.h.289.1		2
20.7	even	4		720.4.f.h.289.2		2
60.23	odd	4		240.4.f.a.49.1		2
60.47	odd	4		240.4.f.a.49.2		2
60.59	even	2		1200.4.a.w.1.1		1
120.53	even	4		960.4.f.j.769.1		2
120.77	even	4		960.4.f.j.769.2		2
120.83	odd	4		960.4.f.i.769.2		2
120.107	odd	4		960.4.f.i.769.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.d.a.49.1	✓	2	15.2	even	4
60.4.d.a.49.2	yes	2	15.8	even	4
180.4.d.b.109.1		2	5.3	odd	4
180.4.d.b.109.2		2	5.2	odd	4
240.4.f.a.49.1		2	60.23	odd	4
240.4.f.a.49.2		2	60.47	odd	4
300.4.a.d.1.1		1	15.14	odd	2
300.4.a.f.1.1		1	3.2	odd	2
720.4.f.h.289.1		2	20.3	even	4
720.4.f.h.289.2		2	20.7	even	4
900.4.a.d.1.1		1	1.1	even	1	trivial
900.4.a.o.1.1		1	5.4	even	2
960.4.f.i.769.1		2	120.107	odd	4
960.4.f.i.769.2		2	120.83	odd	4
960.4.f.j.769.1		2	120.53	even	4
960.4.f.j.769.2		2	120.77	even	4
1200.4.a.q.1.1		1	12.11	even	2
1200.4.a.w.1.1		1	60.59	even	2