Newform orbit 1575.4.a.p

• Label: 1575.4.a.p
• Level: $1575$
• Weight: $4$
• Character orbit: 1575.a
• Self dual: yes
• Analytic conductor: $92.928$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(92.9280082590\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{57}) \)
Defining polynomial:	\( x^{2} - x - 14 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta - 1) q^{2} + (3 \beta + 7) q^{4} - 7 q^{7} + ( - 5 \beta - 41) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

4.27492
−3.27492

−5.27492

0

19.8248

0

0

−7.00000

−62.3746

0

0

1.2

2.27492

0

−2.82475

0

0

−7.00000

−24.6254

0

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(7\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.4.a.p		2
3.b	odd	2	1		525.4.a.n		2
5.b	even	2	1		63.4.a.e		2
15.d	odd	2	1		21.4.a.c	✓	2
15.e	even	4	2		525.4.d.g		4
20.d	odd	2	1		1008.4.a.ba		2
35.c	odd	2	1		441.4.a.r		2
35.i	odd	6	2		441.4.e.p		4
35.j	even	6	2		441.4.e.q		4
60.h	even	2	1		336.4.a.m		2
105.g	even	2	1		147.4.a.i		2
105.o	odd	6	2		147.4.e.l		4
105.p	even	6	2		147.4.e.m		4
120.i	odd	2	1		1344.4.a.bg		2
120.m	even	2	1		1344.4.a.bo		2
420.o	odd	2	1		2352.4.a.bz		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.c	✓	2	15.d	odd	2	1
63.4.a.e		2	5.b	even	2	1
147.4.a.i		2	105.g	even	2	1
147.4.e.l		4	105.o	odd	6	2
147.4.e.m		4	105.p	even	6	2
336.4.a.m		2	60.h	even	2	1
441.4.a.r		2	35.c	odd	2	1
441.4.e.p		4	35.i	odd	6	2
441.4.e.q		4	35.j	even	6	2
525.4.a.n		2	3.b	odd	2	1
525.4.d.g		4	15.e	even	4	2
1008.4.a.ba		2	20.d	odd	2	1
1344.4.a.bg		2	120.i	odd	2	1
1344.4.a.bo		2	120.m	even	2	1
1575.4.a.p		2	1.a	even	1	1	trivial
2352.4.a.bz		2	420.o	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 3T_{2} - 12 \)

\( T_{11}^{2} - 6T_{11} - 1416 \)

\( T_{13}^{2} + 16T_{13} - 1988 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 3T - 12 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 6T - 1416 \)