Newform orbit 1575.3.f.b

• Label: 1575.3.f.b
• Level: $1575$
• Weight: $3$
• Character orbit: 1575.f
• Analytic conductor: $42.916$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1575.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(42.9156416367\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 80 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
449.1	−3.83151				0		10.6805				0			0				2.64575i	−25.5962				0			0
449.2	−3.83151				0		10.6805				0			0			−	2.64575i	−25.5962				0			0
449.3	−3.21234				0		6.31911				0			0			−	2.64575i	−7.44976				0			0
449.4	−3.21234				0		6.31911				0			0				2.64575i	−7.44976				0			0
449.5	−2.96265				0		4.77727				0			0			−	2.64575i	−2.30278				0			0
449.6	−2.96265				0		4.77727				0			0				2.64575i	−2.30278				0			0
449.7	−2.74743				0		3.54835				0			0				2.64575i	1.24087				0			0
449.8	−2.74743				0		3.54835				0			0			−	2.64575i	1.24087				0			0
449.9	−2.62078				0		2.86848				0			0			−	2.64575i	2.96546				0			0
449.10	−2.62078				0		2.86848				0			0				2.64575i	2.96546				0			0
449.11	−1.67508				0		−1.19410				0			0			−	2.64575i	8.70054				0			0
449.12	−1.67508				0		−1.19410				0			0				2.64575i	8.70054				0			0
449.13	−0.982495				0		−3.03470				0			0			−	2.64575i	6.91156				0			0
449.14	−0.982495				0		−3.03470				0			0				2.64575i	6.91156				0			0
449.15	−0.187453				0		−3.96486				0			0				2.64575i	1.49303				0			0
449.16	−0.187453				0		−3.96486				0			0			−	2.64575i	1.49303				0			0
449.17	0.187453				0		−3.96486				0			0			−	2.64575i	−1.49303				0			0
449.18	0.187453				0		−3.96486				0			0				2.64575i	−1.49303				0			0
449.19	0.982495				0		−3.03470				0			0				2.64575i	−6.91156				0			0
449.20	0.982495				0		−3.03470				0			0			−	2.64575i	−6.91156				0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1575.3.f.b		32
3.b	odd	2	1	inner	1575.3.f.b		32
5.b	even	2	1	inner	1575.3.f.b		32
5.c	odd	4	1		315.3.c.a	✓	16
5.c	odd	4	1		1575.3.c.b		16
15.d	odd	2	1	inner	1575.3.f.b		32
15.e	even	4	1		315.3.c.a	✓	16
15.e	even	4	1		1575.3.c.b		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.3.c.a	✓	16	5.c	odd	4	1
315.3.c.a	✓	16	15.e	even	4	1
1575.3.c.b		16	5.c	odd	4	1
1575.3.c.b		16	15.e	even	4	1
1575.3.f.b		32	1.a	even	1	1	trivial
1575.3.f.b		32	3.b	odd	2	1	inner
1575.3.f.b		32	5.b	even	2	1	inner
1575.3.f.b		32	15.d	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^16 - 52*T2^14 + 1096*T2^12 - 12028*T2^10 + 73070*T2^8 - 239132*T2^6 + 372296*T2^4 - 199508*T2^2 + 6561