LMFDB - Newform orbit 1575.4.a.l

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1575,4,Mod(1,1575)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,5,0,17,0,0,-7,45,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$92.9280082590$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q + 5 q^{2} + 17 q^{4} - 7 q^{7} + 45 q^{8} - 12 q^{11} - 30 q^{13} - 35 q^{14} + 89 q^{16} - 134 q^{17} - 92 q^{19} - 60 q^{22} + 112 q^{23} - 150 q^{26} - 119 q^{28} + 58 q^{29} - 224 q^{31} + 85 q^{32}+ \cdots + 245 q^{98}+O(q^{100}) $

q + 5 * q^2 + 17 * q^4 - 7 * q^7 + 45 * q^8 - 12 * q^11 - 30 * q^13 - 35 * q^14 + 89 * q^16 - 134 * q^17 - 92 * q^19 - 60 * q^22 + 112 * q^23 - 150 * q^26 - 119 * q^28 + 58 * q^29 - 224 * q^31 + 85 * q^32 - 670 * q^34 + 146 * q^37 - 460 * q^38 - 18 * q^41 - 340 * q^43 - 204 * q^44 + 560 * q^46 + 208 * q^47 + 49 * q^49 - 510 * q^52 - 754 * q^53 - 315 * q^56 + 290 * q^58 - 380 * q^59 + 718 * q^61 - 1120 * q^62 - 287 * q^64 - 412 * q^67 - 2278 * q^68 + 960 * q^71 - 1066 * q^73 + 730 * q^74 - 1564 * q^76 + 84 * q^77 + 896 * q^79 - 90 * q^82 + 436 * q^83 - 1700 * q^86 - 540 * q^88 + 1038 * q^89 + 210 * q^91 + 1904 * q^92 + 1040 * q^94 + 702 * q^97 + 245 * q^98

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

5.00000

17.0000

−7.00000

45.0000

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.l		1
3.b	odd	2	1		525.4.a.a		1
5.b	even	2	1		315.4.a.a		1
15.d	odd	2	1		105.4.a.b	✓	1
15.e	even	4	2		525.4.d.a		2
35.c	odd	2	1		2205.4.a.b		1
60.h	even	2	1		1680.4.a.u		1
105.g	even	2	1		735.4.a.j		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.4.a.b	✓	1	15.d	odd	2	1
315.4.a.a		1	5.b	even	2	1
525.4.a.a		1	3.b	odd	2	1
525.4.d.a		2	15.e	even	4	2
735.4.a.j		1	105.g	even	2	1
1575.4.a.l		1	1.a	even	1	1	trivial
1680.4.a.u		1	60.h	even	2	1
2205.4.a.b		1	35.c	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} - 5 $

T2 - 5

$ T_{11} + 12 $

T11 + 12

$ T_{13} + 30 $

T13 + 30

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 5 $ T - 5
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 7 $ T + 7
$11$	$ T + 12 $ T + 12
$13$	$ T + 30 $ T + 30
$17$	$ T + 134 $ T + 134
$19$	$ T + 92 $ T + 92
$23$	$ T - 112 $ T - 112
$29$	$ T - 58 $ T - 58
$31$	$ T + 224 $ T + 224
$37$	$ T - 146 $ T - 146
$41$	$ T + 18 $ T + 18
$43$	$ T + 340 $ T + 340
$47$	$ T - 208 $ T - 208
$53$	$ T + 754 $ T + 754
$59$	$ T + 380 $ T + 380
$61$	$ T - 718 $ T - 718
$67$	$ T + 412 $ T + 412
$71$	$ T - 960 $ T - 960
$73$	$ T + 1066 $ T + 1066
$79$	$ T - 896 $ T - 896
$83$	$ T - 436 $ T - 436
$89$	$ T - 1038 $ T - 1038
$97$	$ T - 702 $ T - 702
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Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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