Embedded newform 900.4.a.k.1.1

• Label: 900.4.a.k.1.1
• Level: $900$
• Weight: $4$
• Character: 900.1
• Self dual: yes
• Analytic conductor: $53.102$
• Analytic rank: $0$
• 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:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 300)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+7.00000 q^{7} +54.0000 q^{11} +55.0000 q^{13} +18.0000 q^{17} -25.0000 q^{19} -18.0000 q^{23} +54.0000 q^{29} -271.000 q^{31} -314.000 q^{37} +360.000 q^{41} +163.000 q^{43} +522.000 q^{47} -294.000 q^{49} -36.0000 q^{53} -126.000 q^{59} +47.0000 q^{61} +343.000 q^{67} +1080.00 q^{71} +1054.00 q^{73} +378.000 q^{77} -568.000 q^{79} +1422.00 q^{83} -1440.00 q^{89} +385.000 q^{91} +439.000 q^{97} +O(q^{100})\)

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
			\(7\)	7.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
0.188982	+	0.981981i				\(0.439481\pi\)
\(11\)	54.0000	1.48015	0.740073	−	0.672526i	\(-0.234791\pi\)
			0.740073	+	0.672526i	\(0.234791\pi\)
\(13\)	55.0000	1.17340	0.586702	−	0.809803i	\(-0.300426\pi\)
			0.586702	+	0.809803i	\(0.300426\pi\)
\(17\)	18.0000	0.256802	0.128401	−	0.991722i	\(-0.459015\pi\)
			0.128401	+	0.991722i	\(0.459015\pi\)
\(19\)	−25.0000	−0.301863	−0.150931	−	0.988544i	\(-0.548227\pi\)
			−0.150931	+	0.988544i	\(0.548227\pi\)
\(23\)	−18.0000	−0.163185	−0.0815926	−	0.996666i	\(-0.526001\pi\)
			−0.0815926	+	0.996666i	\(0.526001\pi\)
\(29\)	54.0000	0.345778	0.172889	−	0.984941i	\(-0.444690\pi\)
			0.172889	+	0.984941i	\(0.444690\pi\)
\(31\)	−271.000	−1.57010	−0.785049	−	0.619434i	\(-0.787362\pi\)
			−0.785049	+	0.619434i	\(0.787362\pi\)
\(37\)	−314.000	−1.39517	−0.697585	−	0.716502i	\(-0.745742\pi\)
			−0.697585	+	0.716502i	\(0.745742\pi\)
\(41\)	360.000	1.37128	0.685641	−	0.727940i	\(-0.259522\pi\)
			0.685641	+	0.727940i	\(0.259522\pi\)
\(43\)	163.000	0.578076	0.289038	−	0.957318i	\(-0.406665\pi\)
			0.289038	+	0.957318i	\(0.406665\pi\)
\(47\)	522.000	1.62003	0.810016	−	0.586407i	\(-0.199458\pi\)
			0.810016	+	0.586407i	\(0.199458\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.4.a.k.1.1		1
3.2	odd	2		300.4.a.c.1.1	✓	1
5.2	odd	4		900.4.d.j.649.2		2
5.3	odd	4		900.4.d.j.649.1		2
5.4	even	2		900.4.a.h.1.1		1
12.11	even	2		1200.4.a.y.1.1		1
15.2	even	4		300.4.d.a.49.2		2
15.8	even	4		300.4.d.a.49.1		2
15.14	odd	2		300.4.a.g.1.1	yes	1
60.23	odd	4		1200.4.f.s.49.2		2
60.47	odd	4		1200.4.f.s.49.1		2
60.59	even	2		1200.4.a.m.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.4.a.c.1.1	✓	1	3.2	odd	2
300.4.a.g.1.1	yes	1	15.14	odd	2
300.4.d.a.49.1		2	15.8	even	4
300.4.d.a.49.2		2	15.2	even	4
900.4.a.h.1.1		1	5.4	even	2
900.4.a.k.1.1		1	1.1	even	1	trivial
900.4.d.j.649.1		2	5.3	odd	4
900.4.d.j.649.2		2	5.2	odd	4
1200.4.a.m.1.1		1	60.59	even	2
1200.4.a.y.1.1		1	12.11	even	2
1200.4.f.s.49.1		2	60.47	odd	4
1200.4.f.s.49.2		2	60.23	odd	4