LMFDB - Newform orbit 1350.4.a.t

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1350,4,Mod(1,1350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1350.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,4,0,0,-8,8,0,0,-18] 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:	$79.6525785077$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
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 + 2 q^{2} + 4 q^{4} - 8 q^{7} + 8 q^{8} - 18 q^{11} - 8 q^{13} - 16 q^{14} + 16 q^{16} + 15 q^{17} + 23 q^{19} - 36 q^{22} + 63 q^{23} - 16 q^{26} - 32 q^{28} - 156 q^{29} - 85 q^{31} + 32 q^{32} + 30 q^{34}+ \cdots - 558 q^{98}+O(q^{100}) $

q + 2 * q^2 + 4 * q^4 - 8 * q^7 + 8 * q^8 - 18 * q^11 - 8 * q^13 - 16 * q^14 + 16 * q^16 + 15 * q^17 + 23 * q^19 - 36 * q^22 + 63 * q^23 - 16 * q^26 - 32 * q^28 - 156 * q^29 - 85 * q^31 + 32 * q^32 + 30 * q^34 - 74 * q^37 + 46 * q^38 - 246 * q^41 + 190 * q^43 - 72 * q^44 + 126 * q^46 + 288 * q^47 - 279 * q^49 - 32 * q^52 - 177 * q^53 - 64 * q^56 - 312 * q^58 - 792 * q^59 - 907 * q^61 - 170 * q^62 + 64 * q^64 + 322 * q^67 + 60 * q^68 + 270 * q^71 - 254 * q^73 - 148 * q^74 + 92 * q^76 + 144 * q^77 - 1123 * q^79 - 492 * q^82 - 771 * q^83 + 380 * q^86 - 144 * q^88 + 198 * q^89 + 64 * q^91 + 252 * q^92 + 576 * q^94 + 1192 * q^97 - 558 * 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

2.00000

4.00000

−8.00000

8.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ +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	1350.4.a.t		1
3.b	odd	2	1		1350.4.a.f		1
5.b	even	2	1		270.4.a.b	✓	1
5.c	odd	4	2		1350.4.c.g		2
15.d	odd	2	1		270.4.a.l	yes	1
15.e	even	4	2		1350.4.c.n		2
20.d	odd	2	1		2160.4.a.c		1
45.h	odd	6	2		810.4.e.b		2
45.j	even	6	2		810.4.e.v		2
60.h	even	2	1		2160.4.a.m		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.b	✓	1	5.b	even	2	1
270.4.a.l	yes	1	15.d	odd	2	1
810.4.e.b		2	45.h	odd	6	2
810.4.e.v		2	45.j	even	6	2
1350.4.a.f		1	3.b	odd	2	1
1350.4.a.t		1	1.a	even	1	1	trivial
1350.4.c.g		2	5.c	odd	4	2
1350.4.c.n		2	15.e	even	4	2
2160.4.a.c		1	20.d	odd	2	1
2160.4.a.m		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1350))$:

$ T_{7} + 8 $

T7 + 8

$ T_{11} + 18 $

T11 + 18

$ T_{17} - 15 $

T17 - 15

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 8 $ T + 8
$11$	$ T + 18 $ T + 18
$13$	$ T + 8 $ T + 8
$17$	$ T - 15 $ T - 15
$19$	$ T - 23 $ T - 23
$23$	$ T - 63 $ T - 63
$29$	$ T + 156 $ T + 156
$31$	$ T + 85 $ T + 85
$37$	$ T + 74 $ T + 74
$41$	$ T + 246 $ T + 246
$43$	$ T - 190 $ T - 190
$47$	$ T - 288 $ T - 288
$53$	$ T + 177 $ T + 177
$59$	$ T + 792 $ T + 792
$61$	$ T + 907 $ T + 907
$67$	$ T - 322 $ T - 322
$71$	$ T - 270 $ T - 270
$73$	$ T + 254 $ T + 254
$79$	$ T + 1123 $ T + 1123
$83$	$ T + 771 $ T + 771
$89$	$ T - 198 $ T - 198
$97$	$ T - 1192 $ T - 1192
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Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

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