Embedded newform 75.2.a.b.1.1

• Label: 75.2.a.b.1.1
• Level: $75$
• Weight: $2$
• Character: 75.1
• Self dual: yes
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.598878015160\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	75.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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	0.353553	−	0.935414i	\(-0.384973\pi\)
			0.353553	+	0.935414i	\(0.384973\pi\)
\(3\)	1.00000	0.577350
			\(5\)	0	0
\(7\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
			0.458831	+	0.888523i	\(0.348268\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(37\)	10.0000	1.64399	0.821995	−	0.569495i	\(-0.192861\pi\)
			0.821995	+	0.569495i	\(0.192861\pi\)
\(41\)	10.0000	1.56174	0.780869	−	0.624695i	\(-0.214777\pi\)
			0.780869	+	0.624695i	\(0.214777\pi\)
\(43\)	−4.00000	−0.609994	−0.304997	−	0.952353i	\(-0.598656\pi\)
			−0.304997	+	0.952353i	\(0.598656\pi\)
\(47\)	−8.00000	−1.16692	−0.583460	−	0.812142i	\(-0.698301\pi\)
			−0.583460	+	0.812142i	\(0.698301\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.2.a.b.1.1		1
3.2	odd	2		225.2.a.b.1.1		1
4.3	odd	2		1200.2.a.e.1.1		1
5.2	odd	4		75.2.b.b.49.2		2
5.3	odd	4		75.2.b.b.49.1		2
5.4	even	2		15.2.a.a.1.1	✓	1
7.6	odd	2		3675.2.a.j.1.1		1
8.3	odd	2		4800.2.a.bz.1.1		1
8.5	even	2		4800.2.a.t.1.1		1
11.10	odd	2		9075.2.a.g.1.1		1
12.11	even	2		3600.2.a.u.1.1		1
15.2	even	4		225.2.b.b.199.1		2
15.8	even	4		225.2.b.b.199.2		2
15.14	odd	2		45.2.a.a.1.1		1
20.3	even	4		1200.2.f.h.49.1		2
20.7	even	4		1200.2.f.h.49.2		2
20.19	odd	2		240.2.a.d.1.1		1
35.4	even	6		735.2.i.e.226.1		2
35.9	even	6		735.2.i.e.361.1		2
35.19	odd	6		735.2.i.d.361.1		2
35.24	odd	6		735.2.i.d.226.1		2
35.34	odd	2		735.2.a.c.1.1		1
40.3	even	4		4800.2.f.c.3649.2		2
40.13	odd	4		4800.2.f.bf.3649.1		2
40.19	odd	2		960.2.a.a.1.1		1
40.27	even	4		4800.2.f.c.3649.1		2
40.29	even	2		960.2.a.l.1.1		1
40.37	odd	4		4800.2.f.bf.3649.2		2
45.4	even	6		405.2.e.f.136.1		2
45.14	odd	6		405.2.e.c.136.1		2
45.29	odd	6		405.2.e.c.271.1		2
45.34	even	6		405.2.e.f.271.1		2
55.54	odd	2		1815.2.a.d.1.1		1
60.23	odd	4		3600.2.f.e.2449.2		2
60.47	odd	4		3600.2.f.e.2449.1		2
60.59	even	2		720.2.a.c.1.1		1
65.64	even	2		2535.2.a.j.1.1		1
80.19	odd	4		3840.2.k.r.1921.2		2
80.29	even	4		3840.2.k.m.1921.1		2
80.59	odd	4		3840.2.k.r.1921.1		2
80.69	even	4		3840.2.k.m.1921.2		2
85.84	even	2		4335.2.a.c.1.1		1
95.94	odd	2		5415.2.a.j.1.1		1
105.104	even	2		2205.2.a.i.1.1		1
115.114	odd	2		7935.2.a.d.1.1		1
120.29	odd	2		2880.2.a.y.1.1		1
120.59	even	2		2880.2.a.bc.1.1		1
165.164	even	2		5445.2.a.c.1.1		1
195.194	odd	2		7605.2.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.2.a.a.1.1	✓	1	5.4	even	2
45.2.a.a.1.1		1	15.14	odd	2
75.2.a.b.1.1		1	1.1	even	1	trivial
75.2.b.b.49.1		2	5.3	odd	4
75.2.b.b.49.2		2	5.2	odd	4
225.2.a.b.1.1		1	3.2	odd	2
225.2.b.b.199.1		2	15.2	even	4
225.2.b.b.199.2		2	15.8	even	4
240.2.a.d.1.1		1	20.19	odd	2
405.2.e.c.136.1		2	45.14	odd	6
405.2.e.c.271.1		2	45.29	odd	6
405.2.e.f.136.1		2	45.4	even	6
405.2.e.f.271.1		2	45.34	even	6
720.2.a.c.1.1		1	60.59	even	2
735.2.a.c.1.1		1	35.34	odd	2
735.2.i.d.226.1		2	35.24	odd	6
735.2.i.d.361.1		2	35.19	odd	6
735.2.i.e.226.1		2	35.4	even	6
735.2.i.e.361.1		2	35.9	even	6
960.2.a.a.1.1		1	40.19	odd	2
960.2.a.l.1.1		1	40.29	even	2
1200.2.a.e.1.1		1	4.3	odd	2
1200.2.f.h.49.1		2	20.3	even	4
1200.2.f.h.49.2		2	20.7	even	4
1815.2.a.d.1.1		1	55.54	odd	2
2205.2.a.i.1.1		1	105.104	even	2
2535.2.a.j.1.1		1	65.64	even	2
2880.2.a.y.1.1		1	120.29	odd	2
2880.2.a.bc.1.1		1	120.59	even	2
3600.2.a.u.1.1		1	12.11	even	2
3600.2.f.e.2449.1		2	60.47	odd	4
3600.2.f.e.2449.2		2	60.23	odd	4
3675.2.a.j.1.1		1	7.6	odd	2
3840.2.k.m.1921.1		2	80.29	even	4
3840.2.k.m.1921.2		2	80.69	even	4
3840.2.k.r.1921.1		2	80.59	odd	4
3840.2.k.r.1921.2		2	80.19	odd	4
4335.2.a.c.1.1		1	85.84	even	2
4800.2.a.t.1.1		1	8.5	even	2
4800.2.a.bz.1.1		1	8.3	odd	2
4800.2.f.c.3649.1		2	40.27	even	4
4800.2.f.c.3649.2		2	40.3	even	4
4800.2.f.bf.3649.1		2	40.13	odd	4
4800.2.f.bf.3649.2		2	40.37	odd	4
5415.2.a.j.1.1		1	95.94	odd	2
5445.2.a.c.1.1		1	165.164	even	2
7605.2.a.g.1.1		1	195.194	odd	2
7935.2.a.d.1.1		1	115.114	odd	2
9075.2.a.g.1.1		1	11.10	odd	2