Embedded newform 1575.1.h.b.1126.1

• Label: 1575.1.h.b.1126.1
• Level: $1575$
• Weight: $1$
• Character: 1575.1126
• Self dual: yes
• Analytic conductor: $0.786$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -3, -7, 21
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1575.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.786027394897\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.4725.1
Stark unit:	Root of $x^{4} - 439x^{3} + 801x^{2} - 439x + 1$

Embedding invariants

Embedding label			1126.1
Character	\(\chi\)	\(=\)	1575.1126

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{4} +1.00000 q^{7} +O(q^{10})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(451\)	\(1226\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(3\)	0	0
			\(5\)	0	0
\(7\)	1.00000	1.00000
			\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(13\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(37\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	2.00000	2.00000	1.00000			\(0\)
			1.00000			\(0\)
\(47\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.1.h.b.1126.1		1
3.2	odd	2	CM	1575.1.h.b.1126.1		1
5.2	odd	4		1575.1.e.b.874.2		2
5.3	odd	4		1575.1.e.b.874.1		2
5.4	even	2		63.1.d.a.55.1	✓	1
7.6	odd	2	CM	1575.1.h.b.1126.1		1
15.2	even	4		1575.1.e.b.874.2		2
15.8	even	4		1575.1.e.b.874.1		2
15.14	odd	2		63.1.d.a.55.1	✓	1
20.19	odd	2		1008.1.f.a.433.1		1
21.20	even	2	RM	1575.1.h.b.1126.1		1
35.4	even	6		441.1.m.a.19.1		2
35.9	even	6		441.1.m.a.325.1		2
35.13	even	4		1575.1.e.b.874.1		2
35.19	odd	6		441.1.m.a.325.1		2
35.24	odd	6		441.1.m.a.19.1		2
35.27	even	4		1575.1.e.b.874.2		2
35.34	odd	2		63.1.d.a.55.1	✓	1
45.4	even	6		567.1.l.b.55.1		2
45.14	odd	6		567.1.l.b.55.1		2
45.29	odd	6		567.1.l.b.433.1		2
45.34	even	6		567.1.l.b.433.1		2
60.59	even	2		1008.1.f.a.433.1		1
105.44	odd	6		441.1.m.a.325.1		2
105.59	even	6		441.1.m.a.19.1		2
105.62	odd	4		1575.1.e.b.874.2		2
105.74	odd	6		441.1.m.a.19.1		2
105.83	odd	4		1575.1.e.b.874.1		2
105.89	even	6		441.1.m.a.325.1		2
105.104	even	2		63.1.d.a.55.1	✓	1
140.139	even	2		1008.1.f.a.433.1		1
315.4	even	6		3969.1.k.b.460.1		2
315.34	odd	6		567.1.l.b.433.1		2
315.59	even	6		3969.1.k.b.460.1		2
315.74	odd	6		3969.1.t.c.3106.1		2
315.79	even	6		3969.1.k.b.1648.1		2
315.94	odd	6		3969.1.k.b.460.1		2
315.104	even	6		567.1.l.b.55.1		2
315.124	odd	6		3969.1.k.b.1648.1		2
315.139	odd	6		567.1.l.b.55.1		2
315.149	odd	6		3969.1.t.c.2971.1		2
315.164	even	6		3969.1.t.c.3106.1		2
315.184	even	6		3969.1.t.c.2971.1		2
315.194	even	6		3969.1.t.c.2971.1		2
315.209	even	6		567.1.l.b.433.1		2
315.214	even	6		3969.1.t.c.3106.1		2
315.229	odd	6		3969.1.t.c.2971.1		2
315.254	odd	6		3969.1.k.b.1648.1		2
315.284	odd	6		3969.1.k.b.460.1		2
315.299	even	6		3969.1.k.b.1648.1		2
315.304	odd	6		3969.1.t.c.3106.1		2
420.419	odd	2		1008.1.f.a.433.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.1.d.a.55.1	✓	1	5.4	even	2
63.1.d.a.55.1	✓	1	15.14	odd	2
63.1.d.a.55.1	✓	1	35.34	odd	2
63.1.d.a.55.1	✓	1	105.104	even	2
441.1.m.a.19.1		2	35.4	even	6
441.1.m.a.19.1		2	35.24	odd	6
441.1.m.a.19.1		2	105.59	even	6
441.1.m.a.19.1		2	105.74	odd	6
441.1.m.a.325.1		2	35.9	even	6
441.1.m.a.325.1		2	35.19	odd	6
441.1.m.a.325.1		2	105.44	odd	6
441.1.m.a.325.1		2	105.89	even	6
567.1.l.b.55.1		2	45.4	even	6
567.1.l.b.55.1		2	45.14	odd	6
567.1.l.b.55.1		2	315.104	even	6
567.1.l.b.55.1		2	315.139	odd	6
567.1.l.b.433.1		2	45.29	odd	6
567.1.l.b.433.1		2	45.34	even	6
567.1.l.b.433.1		2	315.34	odd	6
567.1.l.b.433.1		2	315.209	even	6
1008.1.f.a.433.1		1	20.19	odd	2
1008.1.f.a.433.1		1	60.59	even	2
1008.1.f.a.433.1		1	140.139	even	2
1008.1.f.a.433.1		1	420.419	odd	2
1575.1.e.b.874.1		2	5.3	odd	4
1575.1.e.b.874.1		2	15.8	even	4
1575.1.e.b.874.1		2	35.13	even	4
1575.1.e.b.874.1		2	105.83	odd	4
1575.1.e.b.874.2		2	5.2	odd	4
1575.1.e.b.874.2		2	15.2	even	4
1575.1.e.b.874.2		2	35.27	even	4
1575.1.e.b.874.2		2	105.62	odd	4
1575.1.h.b.1126.1		1	1.1	even	1	trivial
1575.1.h.b.1126.1		1	3.2	odd	2	CM
1575.1.h.b.1126.1		1	7.6	odd	2	CM
1575.1.h.b.1126.1		1	21.20	even	2	RM
3969.1.k.b.460.1		2	315.4	even	6
3969.1.k.b.460.1		2	315.59	even	6
3969.1.k.b.460.1		2	315.94	odd	6
3969.1.k.b.460.1		2	315.284	odd	6
3969.1.k.b.1648.1		2	315.79	even	6
3969.1.k.b.1648.1		2	315.124	odd	6
3969.1.k.b.1648.1		2	315.254	odd	6
3969.1.k.b.1648.1		2	315.299	even	6
3969.1.t.c.2971.1		2	315.149	odd	6
3969.1.t.c.2971.1		2	315.184	even	6
3969.1.t.c.2971.1		2	315.194	even	6
3969.1.t.c.2971.1		2	315.229	odd	6
3969.1.t.c.3106.1		2	315.74	odd	6
3969.1.t.c.3106.1		2	315.164	even	6
3969.1.t.c.3106.1		2	315.214	even	6
3969.1.t.c.3106.1		2	315.304	odd	6