LMFDB - Newform orbit 1350.4.a.w

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:	$0$
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} + 13 q^{7} + 8 q^{8}+O(q^{10}) $

$ q + 2 q^{2} + 4 q^{4} + 13 q^{7} + 8 q^{8} + 30 q^{11} + 61 q^{13} + 26 q^{14} + 16 q^{16} + 12 q^{17} - 49 q^{19} + 60 q^{22} + 18 q^{23} + 122 q^{26} + 52 q^{28} + 186 q^{29} - 160 q^{31} + 32 q^{32} + 24 q^{34} + 91 q^{37} - 98 q^{38} - 378 q^{41} + 268 q^{43} + 120 q^{44} + 36 q^{46} + 144 q^{47} - 174 q^{49} + 244 q^{52} + 570 q^{53} + 104 q^{56} + 372 q^{58} - 204 q^{59} - 877 q^{61} - 320 q^{62} + 64 q^{64} + 187 q^{67} + 48 q^{68} + 606 q^{71} - 431 q^{73} + 182 q^{74} - 196 q^{76} + 390 q^{77} + 1151 q^{79} - 756 q^{82} + 102 q^{83} + 536 q^{86} + 240 q^{88} - 984 q^{89} + 793 q^{91} + 72 q^{92} + 288 q^{94} + 265 q^{97} - 348 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

13.0000

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.w		1
3.b	odd	2	1		1350.4.a.i		1
5.b	even	2	1		270.4.a.d	✓	1
5.c	odd	4	2		1350.4.c.o		2
15.d	odd	2	1		270.4.a.h	yes	1
15.e	even	4	2		1350.4.c.f		2
20.d	odd	2	1		2160.4.a.q		1
45.h	odd	6	2		810.4.e.j		2
45.j	even	6	2		810.4.e.r		2
60.h	even	2	1		2160.4.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.d	✓	1	5.b	even	2	1
270.4.a.h	yes	1	15.d	odd	2	1
810.4.e.j		2	45.h	odd	6	2
810.4.e.r		2	45.j	even	6	2
1350.4.a.i		1	3.b	odd	2	1
1350.4.a.w		1	1.a	even	1	1	trivial
1350.4.c.f		2	15.e	even	4	2
1350.4.c.o		2	5.c	odd	4	2
2160.4.a.g		1	60.h	even	2	1
2160.4.a.q		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} - 13 $

$ T_{11} - 30 $

$ T_{17} - 12 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 13 $ T - 13
$11$	$ T - 30 $ T - 30
$13$	$ T - 61 $ T - 61
$17$	$ T - 12 $ T - 12
$19$	$ T + 49 $ T + 49
$23$	$ T - 18 $ T - 18
$29$	$ T - 186 $ T - 186
$31$	$ T + 160 $ T + 160
$37$	$ T - 91 $ T - 91
$41$	$ T + 378 $ T + 378
$43$	$ T - 268 $ T - 268
$47$	$ T - 144 $ T - 144
$53$	$ T - 570 $ T - 570
$59$	$ T + 204 $ T + 204
$61$	$ T + 877 $ T + 877
$67$	$ T - 187 $ T - 187
$71$	$ T - 606 $ T - 606
$73$	$ T + 431 $ T + 431
$79$	$ T - 1151 $ T - 1151
$83$	$ T - 102 $ T - 102
$89$	$ T + 984 $ T + 984
$97$	$ T - 265 $ T - 265
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

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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