Embedded newform 90.2.a.c.1.1

• Label: 90.2.a.c.1.1
• Level: $90$
• Weight: $2$
• Character: 90.1
• Self dual: yes
• Analytic conductor: $0.719$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.718653618192\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	90.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	0	0
\(5\)	1.00000	0.447214
			\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
−0.755929	+	0.654654i				\(0.772814\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−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	\(0.651732\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	6.00000	1.11417	0.557086	−	0.830455i	\(-0.311919\pi\)
			0.557086	+	0.830455i	\(0.311919\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	2.00000	0.328798	0.164399	−	0.986394i	\(-0.447432\pi\)
			0.164399	+	0.986394i	\(0.447432\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.2.a.c.1.1		1
3.2	odd	2		30.2.a.a.1.1	✓	1
4.3	odd	2		720.2.a.j.1.1		1
5.2	odd	4		450.2.c.b.199.2		2
5.3	odd	4		450.2.c.b.199.1		2
5.4	even	2		450.2.a.d.1.1		1
7.6	odd	2		4410.2.a.z.1.1		1
8.3	odd	2		2880.2.a.q.1.1		1
8.5	even	2		2880.2.a.a.1.1		1
9.2	odd	6		810.2.e.l.271.1		2
9.4	even	3		810.2.e.b.541.1		2
9.5	odd	6		810.2.e.l.541.1		2
9.7	even	3		810.2.e.b.271.1		2
12.11	even	2		240.2.a.b.1.1		1
15.2	even	4		150.2.c.a.49.1		2
15.8	even	4		150.2.c.a.49.2		2
15.14	odd	2		150.2.a.b.1.1		1
20.3	even	4		3600.2.f.i.2449.1		2
20.7	even	4		3600.2.f.i.2449.2		2
20.19	odd	2		3600.2.a.f.1.1		1
21.2	odd	6		1470.2.i.o.361.1		2
21.5	even	6		1470.2.i.q.361.1		2
21.11	odd	6		1470.2.i.o.961.1		2
21.17	even	6		1470.2.i.q.961.1		2
21.20	even	2		1470.2.a.d.1.1		1
24.5	odd	2		960.2.a.e.1.1		1
24.11	even	2		960.2.a.p.1.1		1
33.32	even	2		3630.2.a.w.1.1		1
39.5	even	4		5070.2.b.k.1351.2		2
39.8	even	4		5070.2.b.k.1351.1		2
39.38	odd	2		5070.2.a.w.1.1		1
48.5	odd	4		3840.2.k.y.1921.1		2
48.11	even	4		3840.2.k.f.1921.2		2
48.29	odd	4		3840.2.k.y.1921.2		2
48.35	even	4		3840.2.k.f.1921.1		2
51.50	odd	2		8670.2.a.g.1.1		1
60.23	odd	4		1200.2.f.e.49.1		2
60.47	odd	4		1200.2.f.e.49.2		2
60.59	even	2		1200.2.a.k.1.1		1
105.104	even	2		7350.2.a.ct.1.1		1
120.29	odd	2		4800.2.a.cq.1.1		1
120.53	even	4		4800.2.f.p.3649.1		2
120.59	even	2		4800.2.a.d.1.1		1
120.77	even	4		4800.2.f.p.3649.2		2
120.83	odd	4		4800.2.f.w.3649.2		2
120.107	odd	4		4800.2.f.w.3649.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	3.2	odd	2
90.2.a.c.1.1		1	1.1	even	1	trivial
150.2.a.b.1.1		1	15.14	odd	2
150.2.c.a.49.1		2	15.2	even	4
150.2.c.a.49.2		2	15.8	even	4
240.2.a.b.1.1		1	12.11	even	2
450.2.a.d.1.1		1	5.4	even	2
450.2.c.b.199.1		2	5.3	odd	4
450.2.c.b.199.2		2	5.2	odd	4
720.2.a.j.1.1		1	4.3	odd	2
810.2.e.b.271.1		2	9.7	even	3
810.2.e.b.541.1		2	9.4	even	3
810.2.e.l.271.1		2	9.2	odd	6
810.2.e.l.541.1		2	9.5	odd	6
960.2.a.e.1.1		1	24.5	odd	2
960.2.a.p.1.1		1	24.11	even	2
1200.2.a.k.1.1		1	60.59	even	2
1200.2.f.e.49.1		2	60.23	odd	4
1200.2.f.e.49.2		2	60.47	odd	4
1470.2.a.d.1.1		1	21.20	even	2
1470.2.i.o.361.1		2	21.2	odd	6
1470.2.i.o.961.1		2	21.11	odd	6
1470.2.i.q.361.1		2	21.5	even	6
1470.2.i.q.961.1		2	21.17	even	6
2880.2.a.a.1.1		1	8.5	even	2
2880.2.a.q.1.1		1	8.3	odd	2
3600.2.a.f.1.1		1	20.19	odd	2
3600.2.f.i.2449.1		2	20.3	even	4
3600.2.f.i.2449.2		2	20.7	even	4
3630.2.a.w.1.1		1	33.32	even	2
3840.2.k.f.1921.1		2	48.35	even	4
3840.2.k.f.1921.2		2	48.11	even	4
3840.2.k.y.1921.1		2	48.5	odd	4
3840.2.k.y.1921.2		2	48.29	odd	4
4410.2.a.z.1.1		1	7.6	odd	2
4800.2.a.d.1.1		1	120.59	even	2
4800.2.a.cq.1.1		1	120.29	odd	2
4800.2.f.p.3649.1		2	120.53	even	4
4800.2.f.p.3649.2		2	120.77	even	4
4800.2.f.w.3649.1		2	120.107	odd	4
4800.2.f.w.3649.2		2	120.83	odd	4
5070.2.a.w.1.1		1	39.38	odd	2
5070.2.b.k.1351.1		2	39.8	even	4
5070.2.b.k.1351.2		2	39.5	even	4
7350.2.a.ct.1.1		1	105.104	even	2
8670.2.a.g.1.1		1	51.50	odd	2