Embedded newform 900.2.a.b.1.1

• Label: 900.2.a.b.1.1
• Level: $900$
• Weight: $2$
• Character: 900.1
• Self dual: yes
• Analytic conductor: $7.187$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{7} -2.00000 q^{13} -6.00000 q^{17} -4.00000 q^{19} +6.00000 q^{23} -6.00000 q^{29} -4.00000 q^{31} -2.00000 q^{37} -6.00000 q^{41} +10.0000 q^{43} -6.00000 q^{47} -3.00000 q^{49} -6.00000 q^{53} -12.0000 q^{59} +2.00000 q^{61} -2.00000 q^{67} +12.0000 q^{71} -2.00000 q^{73} +8.00000 q^{79} +6.00000 q^{83} +6.00000 q^{89} +4.00000 q^{91} -2.00000 q^{97} +O(q^{100})\)

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
			\(7\)	−2.00000	−0.755929	−0.377964	−	0.925820i	\(-0.623376\pi\)
−0.377964	+	0.925820i				\(0.623376\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	10.0000	1.52499	0.762493	−	0.646997i	\(-0.223975\pi\)
			0.762493	+	0.646997i	\(0.223975\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.2.a.b.1.1		1
3.2	odd	2		100.2.a.a.1.1		1
4.3	odd	2		3600.2.a.be.1.1		1
5.2	odd	4		900.2.d.c.649.1		2
5.3	odd	4		900.2.d.c.649.2		2
5.4	even	2		180.2.a.a.1.1		1
12.11	even	2		400.2.a.c.1.1		1
15.2	even	4		100.2.c.a.49.1		2
15.8	even	4		100.2.c.a.49.2		2
15.14	odd	2		20.2.a.a.1.1	✓	1
20.3	even	4		3600.2.f.j.2449.1		2
20.7	even	4		3600.2.f.j.2449.2		2
20.19	odd	2		720.2.a.h.1.1		1
21.20	even	2		4900.2.a.e.1.1		1
24.5	odd	2		1600.2.a.c.1.1		1
24.11	even	2		1600.2.a.w.1.1		1
35.34	odd	2		8820.2.a.g.1.1		1
40.19	odd	2		2880.2.a.f.1.1		1
40.29	even	2		2880.2.a.m.1.1		1
45.4	even	6		1620.2.i.b.541.1		2
45.14	odd	6		1620.2.i.h.541.1		2
45.29	odd	6		1620.2.i.h.1081.1		2
45.34	even	6		1620.2.i.b.1081.1		2
60.23	odd	4		400.2.c.b.49.1		2
60.47	odd	4		400.2.c.b.49.2		2
60.59	even	2		80.2.a.b.1.1		1
105.44	odd	6		980.2.i.i.361.1		2
105.59	even	6		980.2.i.c.961.1		2
105.62	odd	4		4900.2.e.f.2549.2		2
105.74	odd	6		980.2.i.i.961.1		2
105.83	odd	4		4900.2.e.f.2549.1		2
105.89	even	6		980.2.i.c.361.1		2
105.104	even	2		980.2.a.h.1.1		1
120.29	odd	2		320.2.a.f.1.1		1
120.53	even	4		1600.2.c.d.449.1		2
120.59	even	2		320.2.a.a.1.1		1
120.77	even	4		1600.2.c.d.449.2		2
120.83	odd	4		1600.2.c.e.449.2		2
120.107	odd	4		1600.2.c.e.449.1		2
165.164	even	2		2420.2.a.a.1.1		1
195.44	even	4		3380.2.f.b.3041.2		2
195.164	even	4		3380.2.f.b.3041.1		2
195.194	odd	2		3380.2.a.c.1.1		1
240.29	odd	4		1280.2.d.c.641.1		2
240.59	even	4		1280.2.d.g.641.1		2
240.149	odd	4		1280.2.d.c.641.2		2
240.179	even	4		1280.2.d.g.641.2		2
255.89	odd	4		5780.2.c.a.5201.2		2
255.149	odd	4		5780.2.c.a.5201.1		2
255.254	odd	2		5780.2.a.f.1.1		1
285.284	even	2		7220.2.a.f.1.1		1
420.419	odd	2		3920.2.a.h.1.1		1
660.659	odd	2		9680.2.a.ba.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.a.a.1.1	✓	1	15.14	odd	2
80.2.a.b.1.1		1	60.59	even	2
100.2.a.a.1.1		1	3.2	odd	2
100.2.c.a.49.1		2	15.2	even	4
100.2.c.a.49.2		2	15.8	even	4
180.2.a.a.1.1		1	5.4	even	2
320.2.a.a.1.1		1	120.59	even	2
320.2.a.f.1.1		1	120.29	odd	2
400.2.a.c.1.1		1	12.11	even	2
400.2.c.b.49.1		2	60.23	odd	4
400.2.c.b.49.2		2	60.47	odd	4
720.2.a.h.1.1		1	20.19	odd	2
900.2.a.b.1.1		1	1.1	even	1	trivial
900.2.d.c.649.1		2	5.2	odd	4
900.2.d.c.649.2		2	5.3	odd	4
980.2.a.h.1.1		1	105.104	even	2
980.2.i.c.361.1		2	105.89	even	6
980.2.i.c.961.1		2	105.59	even	6
980.2.i.i.361.1		2	105.44	odd	6
980.2.i.i.961.1		2	105.74	odd	6
1280.2.d.c.641.1		2	240.29	odd	4
1280.2.d.c.641.2		2	240.149	odd	4
1280.2.d.g.641.1		2	240.59	even	4
1280.2.d.g.641.2		2	240.179	even	4
1600.2.a.c.1.1		1	24.5	odd	2
1600.2.a.w.1.1		1	24.11	even	2
1600.2.c.d.449.1		2	120.53	even	4
1600.2.c.d.449.2		2	120.77	even	4
1600.2.c.e.449.1		2	120.107	odd	4
1600.2.c.e.449.2		2	120.83	odd	4
1620.2.i.b.541.1		2	45.4	even	6
1620.2.i.b.1081.1		2	45.34	even	6
1620.2.i.h.541.1		2	45.14	odd	6
1620.2.i.h.1081.1		2	45.29	odd	6
2420.2.a.a.1.1		1	165.164	even	2
2880.2.a.f.1.1		1	40.19	odd	2
2880.2.a.m.1.1		1	40.29	even	2
3380.2.a.c.1.1		1	195.194	odd	2
3380.2.f.b.3041.1		2	195.164	even	4
3380.2.f.b.3041.2		2	195.44	even	4
3600.2.a.be.1.1		1	4.3	odd	2
3600.2.f.j.2449.1		2	20.3	even	4
3600.2.f.j.2449.2		2	20.7	even	4
3920.2.a.h.1.1		1	420.419	odd	2
4900.2.a.e.1.1		1	21.20	even	2
4900.2.e.f.2549.1		2	105.83	odd	4
4900.2.e.f.2549.2		2	105.62	odd	4
5780.2.a.f.1.1		1	255.254	odd	2
5780.2.c.a.5201.1		2	255.149	odd	4
5780.2.c.a.5201.2		2	255.89	odd	4
7220.2.a.f.1.1		1	285.284	even	2
8820.2.a.g.1.1		1	35.34	odd	2
9680.2.a.ba.1.1		1	660.659	odd	2