Embedded newform 4800.2.a.d.1.1

• Label: 4800.2.a.d.1.1
• Level: $4800$
• Weight: $2$
• Character: 4800.1
• Self dual: yes
• Analytic conductor: $38.328$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(38.3281929702\)
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	\(\chi\)	\(=\)	4800.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} +2.00000 q^{13} -6.00000 q^{17} -4.00000 q^{19} +4.00000 q^{21} -1.00000 q^{27} +6.00000 q^{29} -8.00000 q^{31} +2.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +9.00000 q^{49} +6.00000 q^{51} -6.00000 q^{53} +4.00000 q^{57} +10.0000 q^{61} -4.00000 q^{63} +4.00000 q^{67} -2.00000 q^{73} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -6.00000 q^{87} +18.0000 q^{89} -8.00000 q^{91} +8.00000 q^{93} -2.00000 q^{97} +O(q^{100})\)

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
			\(3\)	−1.00000	−0.577350
\(5\)	0	0
			\(7\)	−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	\(-0.755135\pi\)
			−0.718421	+	0.695608i	\(0.755135\pi\)
\(37\)	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	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	4.00000	0.609994	0.304997	−	0.952353i	\(-0.401344\pi\)
			0.304997	+	0.952353i	\(0.401344\pi\)
\(47\)	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	4800.2.a.d.1.1		1
4.3	odd	2		4800.2.a.cq.1.1		1
5.2	odd	4		4800.2.f.w.3649.2		2
5.3	odd	4		4800.2.f.w.3649.1		2
5.4	even	2		960.2.a.p.1.1		1
8.3	odd	2		150.2.a.b.1.1		1
8.5	even	2		1200.2.a.k.1.1		1
15.14	odd	2		2880.2.a.q.1.1		1
20.3	even	4		4800.2.f.p.3649.2		2
20.7	even	4		4800.2.f.p.3649.1		2
20.19	odd	2		960.2.a.e.1.1		1
24.5	odd	2		3600.2.a.f.1.1		1
24.11	even	2		450.2.a.d.1.1		1
40.3	even	4		150.2.c.a.49.1		2
40.13	odd	4		1200.2.f.e.49.2		2
40.19	odd	2		30.2.a.a.1.1	✓	1
40.27	even	4		150.2.c.a.49.2		2
40.29	even	2		240.2.a.b.1.1		1
40.37	odd	4		1200.2.f.e.49.1		2
56.27	even	2		7350.2.a.ct.1.1		1
60.59	even	2		2880.2.a.a.1.1		1
80.19	odd	4		3840.2.k.y.1921.1		2
80.29	even	4		3840.2.k.f.1921.2		2
80.59	odd	4		3840.2.k.y.1921.2		2
80.69	even	4		3840.2.k.f.1921.1		2
120.29	odd	2		720.2.a.j.1.1		1
120.53	even	4		3600.2.f.i.2449.2		2
120.59	even	2		90.2.a.c.1.1		1
120.77	even	4		3600.2.f.i.2449.1		2
120.83	odd	4		450.2.c.b.199.2		2
120.107	odd	4		450.2.c.b.199.1		2
280.19	even	6		1470.2.i.q.361.1		2
280.59	even	6		1470.2.i.q.961.1		2
280.139	even	2		1470.2.a.d.1.1		1
280.179	odd	6		1470.2.i.o.961.1		2
280.219	odd	6		1470.2.i.o.361.1		2
360.59	even	6		810.2.e.b.541.1		2
360.139	odd	6		810.2.e.l.541.1		2
360.259	odd	6		810.2.e.l.271.1		2
360.299	even	6		810.2.e.b.271.1		2
440.219	even	2		3630.2.a.w.1.1		1
520.99	even	4		5070.2.b.k.1351.1		2
520.259	odd	2		5070.2.a.w.1.1		1
520.499	even	4		5070.2.b.k.1351.2		2
680.339	odd	2		8670.2.a.g.1.1		1
840.419	odd	2		4410.2.a.z.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	40.19	odd	2
90.2.a.c.1.1		1	120.59	even	2
150.2.a.b.1.1		1	8.3	odd	2
150.2.c.a.49.1		2	40.3	even	4
150.2.c.a.49.2		2	40.27	even	4
240.2.a.b.1.1		1	40.29	even	2
450.2.a.d.1.1		1	24.11	even	2
450.2.c.b.199.1		2	120.107	odd	4
450.2.c.b.199.2		2	120.83	odd	4
720.2.a.j.1.1		1	120.29	odd	2
810.2.e.b.271.1		2	360.299	even	6
810.2.e.b.541.1		2	360.59	even	6
810.2.e.l.271.1		2	360.259	odd	6
810.2.e.l.541.1		2	360.139	odd	6
960.2.a.e.1.1		1	20.19	odd	2
960.2.a.p.1.1		1	5.4	even	2
1200.2.a.k.1.1		1	8.5	even	2
1200.2.f.e.49.1		2	40.37	odd	4
1200.2.f.e.49.2		2	40.13	odd	4
1470.2.a.d.1.1		1	280.139	even	2
1470.2.i.o.361.1		2	280.219	odd	6
1470.2.i.o.961.1		2	280.179	odd	6
1470.2.i.q.361.1		2	280.19	even	6
1470.2.i.q.961.1		2	280.59	even	6
2880.2.a.a.1.1		1	60.59	even	2
2880.2.a.q.1.1		1	15.14	odd	2
3600.2.a.f.1.1		1	24.5	odd	2
3600.2.f.i.2449.1		2	120.77	even	4
3600.2.f.i.2449.2		2	120.53	even	4
3630.2.a.w.1.1		1	440.219	even	2
3840.2.k.f.1921.1		2	80.69	even	4
3840.2.k.f.1921.2		2	80.29	even	4
3840.2.k.y.1921.1		2	80.19	odd	4
3840.2.k.y.1921.2		2	80.59	odd	4
4410.2.a.z.1.1		1	840.419	odd	2
4800.2.a.d.1.1		1	1.1	even	1	trivial
4800.2.a.cq.1.1		1	4.3	odd	2
4800.2.f.p.3649.1		2	20.7	even	4
4800.2.f.p.3649.2		2	20.3	even	4
4800.2.f.w.3649.1		2	5.3	odd	4
4800.2.f.w.3649.2		2	5.2	odd	4
5070.2.a.w.1.1		1	520.259	odd	2
5070.2.b.k.1351.1		2	520.99	even	4
5070.2.b.k.1351.2		2	520.499	even	4
7350.2.a.ct.1.1		1	56.27	even	2
8670.2.a.g.1.1		1	680.339	odd	2