LMFDB - Newform orbit 1350.4.a.l

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} + 22 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} + 22 q^{7} - 8 q^{8} - 12 q^{11} - 38 q^{13} - 44 q^{14} + 16 q^{16} + 105 q^{17} - 157 q^{19} + 24 q^{22} + 117 q^{23} + 76 q^{26} + 88 q^{28} + 66 q^{29} - 25 q^{31} - 32 q^{32} - 210 q^{34} - 314 q^{37} + 314 q^{38} - 504 q^{41} - 380 q^{43} - 48 q^{44} - 234 q^{46} + 252 q^{47} + 141 q^{49} - 152 q^{52} - 3 q^{53} - 176 q^{56} - 132 q^{58} - 318 q^{59} + 293 q^{61} + 50 q^{62} + 64 q^{64} + 322 q^{67} + 420 q^{68} - 120 q^{71} - 44 q^{73} + 628 q^{74} - 628 q^{76} - 264 q^{77} + 917 q^{79} + 1008 q^{82} - 309 q^{83} + 760 q^{86} + 96 q^{88} + 1272 q^{89} - 836 q^{91} + 468 q^{92} - 504 q^{94} - 1328 q^{97} - 282 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

22.0000

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

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.c	✓	1	15.d	odd	2	1
270.4.a.g	yes	1	5.b	even	2	1
810.4.e.k		2	45.j	even	6	2
810.4.e.s		2	45.h	odd	6	2
1350.4.a.l		1	1.a	even	1	1	trivial
1350.4.a.z		1	3.b	odd	2	1
1350.4.c.i		2	5.c	odd	4	2
1350.4.c.l		2	15.e	even	4	2
2160.4.a.i		1	20.d	odd	2	1
2160.4.a.r		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} - 22 $

$ T_{11} + 12 $

$ T_{17} - 105 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 22 $ T - 22
$11$	$ T + 12 $ T + 12
$13$	$ T + 38 $ T + 38
$17$	$ T - 105 $ T - 105
$19$	$ T + 157 $ T + 157
$23$	$ T - 117 $ T - 117
$29$	$ T - 66 $ T - 66
$31$	$ T + 25 $ T + 25
$37$	$ T + 314 $ T + 314
$41$	$ T + 504 $ T + 504
$43$	$ T + 380 $ T + 380
$47$	$ T - 252 $ T - 252
$53$	$ T + 3 $ T + 3
$59$	$ T + 318 $ T + 318
$61$	$ T - 293 $ T - 293
$67$	$ T - 322 $ T - 322
$71$	$ T + 120 $ T + 120
$73$	$ T + 44 $ T + 44
$79$	$ T - 917 $ T - 917
$83$	$ T + 309 $ T + 309
$89$	$ T - 1272 $ T - 1272
$97$	$ T + 1328 $ T + 1328
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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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