LMFDB - Newform orbit 1350.4.a.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(1,1350)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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} - 14 q^{7} + 8 q^{8}+O(q^{10}) $

$ q + 2 q^{2} + 4 q^{4} - 14 q^{7} + 8 q^{8} + 3 q^{11} - 47 q^{13} - 28 q^{14} + 16 q^{16} + 39 q^{17} + 32 q^{19} + 6 q^{22} + 99 q^{23} - 94 q^{26} - 56 q^{28} + 51 q^{29} + 83 q^{31} + 32 q^{32} + 78 q^{34} - 314 q^{37} + 64 q^{38} - 108 q^{41} - 299 q^{43} + 12 q^{44} + 198 q^{46} - 531 q^{47} - 147 q^{49} - 188 q^{52} - 564 q^{53} - 112 q^{56} + 102 q^{58} + 12 q^{59} + 230 q^{61} + 166 q^{62} + 64 q^{64} + 268 q^{67} + 156 q^{68} + 120 q^{71} - 1106 q^{73} - 628 q^{74} + 128 q^{76} - 42 q^{77} - 739 q^{79} - 216 q^{82} - 1086 q^{83} - 598 q^{86} + 24 q^{88} - 120 q^{89} + 658 q^{91} + 396 q^{92} - 1062 q^{94} + 1642 q^{97} - 294 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

−14.0000

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.r		1
3.b	odd	2	1		1350.4.a.e		1
5.b	even	2	1		270.4.a.f	✓	1
5.c	odd	4	2		1350.4.c.k		2
15.d	odd	2	1		270.4.a.j	yes	1
15.e	even	4	2		1350.4.c.j		2
20.d	odd	2	1		2160.4.a.l		1
45.h	odd	6	2		810.4.e.f		2
45.j	even	6	2		810.4.e.n		2
60.h	even	2	1		2160.4.a.b		1

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

Hecke kernels

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

$ T_{7} + 14 $

$ T_{11} - 3 $

$ T_{17} - 39 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 14 $ T + 14
$11$	$ T - 3 $ T - 3
$13$	$ T + 47 $ T + 47
$17$	$ T - 39 $ T - 39
$19$	$ T - 32 $ T - 32
$23$	$ T - 99 $ T - 99
$29$	$ T - 51 $ T - 51
$31$	$ T - 83 $ T - 83
$37$	$ T + 314 $ T + 314
$41$	$ T + 108 $ T + 108
$43$	$ T + 299 $ T + 299
$47$	$ T + 531 $ T + 531
$53$	$ T + 564 $ T + 564
$59$	$ T - 12 $ T - 12
$61$	$ T - 230 $ T - 230
$67$	$ T - 268 $ T - 268
$71$	$ T - 120 $ T - 120
$73$	$ T + 1106 $ T + 1106
$79$	$ T + 739 $ T + 739
$83$	$ T + 1086 $ T + 1086
$89$	$ T + 120 $ T + 120
$97$	$ T - 1642 $ T - 1642
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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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