Newform orbit 1575.1.h.b

• Label: 1575.1.h.b
• Level: $1575$
• Weight: $1$
• Character orbit: 1575.h
• 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 \(\Q(\sqrt{-90 +10 \sqrt{-3}})\)
Stark unit:	Root of $x^{4} - 439x^{3} + 801x^{2} - 439x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{4} + q^{7} + q^{16} - q^{28} + 2 q^{37} + 2 q^{43} + q^{49} - q^{64} - 2 q^{67} + 2 q^{79}+O(q^{100}) \)

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1126.1

0

0

0

−1.00000

0

0

1.00000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
21.c	even	2	1	RM by \(\Q(\sqrt{21}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.1.h.b		1
3.b	odd	2	1	CM	1575.1.h.b		1
5.b	even	2	1		63.1.d.a	✓	1
5.c	odd	4	2		1575.1.e.b		2
7.b	odd	2	1	CM	1575.1.h.b		1
15.d	odd	2	1		63.1.d.a	✓	1
15.e	even	4	2		1575.1.e.b		2
20.d	odd	2	1		1008.1.f.a		1
21.c	even	2	1	RM	1575.1.h.b		1
35.c	odd	2	1		63.1.d.a	✓	1
35.f	even	4	2		1575.1.e.b		2
35.i	odd	6	2		441.1.m.a		2
35.j	even	6	2		441.1.m.a		2
45.h	odd	6	2		567.1.l.b		2
45.j	even	6	2		567.1.l.b		2
60.h	even	2	1		1008.1.f.a		1
105.g	even	2	1		63.1.d.a	✓	1
105.k	odd	4	2		1575.1.e.b		2
105.o	odd	6	2		441.1.m.a		2
105.p	even	6	2		441.1.m.a		2
140.c	even	2	1		1008.1.f.a		1
315.q	odd	6	2		3969.1.t.c		2
315.r	even	6	2		3969.1.t.c		2
315.u	even	6	2		3969.1.k.b		2
315.v	odd	6	2		3969.1.k.b		2
315.z	even	6	2		567.1.l.b		2
315.bg	odd	6	2		567.1.l.b		2
315.bn	odd	6	2		3969.1.k.b		2
315.bo	even	6	2		3969.1.k.b		2
315.bq	even	6	2		3969.1.t.c		2
315.br	odd	6	2		3969.1.t.c		2
420.o	odd	2	1		1008.1.f.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.1.d.a	✓	1	5.b	even	2	1
63.1.d.a	✓	1	15.d	odd	2	1
63.1.d.a	✓	1	35.c	odd	2	1
63.1.d.a	✓	1	105.g	even	2	1
441.1.m.a		2	35.i	odd	6	2
441.1.m.a		2	35.j	even	6	2
441.1.m.a		2	105.o	odd	6	2
441.1.m.a		2	105.p	even	6	2
567.1.l.b		2	45.h	odd	6	2
567.1.l.b		2	45.j	even	6	2
567.1.l.b		2	315.z	even	6	2
567.1.l.b		2	315.bg	odd	6	2
1008.1.f.a		1	20.d	odd	2	1
1008.1.f.a		1	60.h	even	2	1
1008.1.f.a		1	140.c	even	2	1
1008.1.f.a		1	420.o	odd	2	1
1575.1.e.b		2	5.c	odd	4	2
1575.1.e.b		2	15.e	even	4	2
1575.1.e.b		2	35.f	even	4	2
1575.1.e.b		2	105.k	odd	4	2
1575.1.h.b		1	1.a	even	1	1	trivial
1575.1.h.b		1	3.b	odd	2	1	CM
1575.1.h.b		1	7.b	odd	2	1	CM
1575.1.h.b		1	21.c	even	2	1	RM
3969.1.k.b		2	315.u	even	6	2
3969.1.k.b		2	315.v	odd	6	2
3969.1.k.b		2	315.bn	odd	6	2
3969.1.k.b		2	315.bo	even	6	2
3969.1.t.c		2	315.q	odd	6	2
3969.1.t.c		2	315.r	even	6	2
3969.1.t.c		2	315.bq	even	6	2
3969.1.t.c		2	315.br	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2} \)

\( T_{37} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T - 1 \)
$11$	\( T \)