LMFDB - Newform orbit 300.4.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,4,Mod(1,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("300.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	300.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:	$17.7005730017$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
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 - 3 q^{3} + 22 q^{7} + 9 q^{9} - 14 q^{11} - 30 q^{13} + 62 q^{17} - 120 q^{19} - 66 q^{21} + 188 q^{23} - 27 q^{27} + 96 q^{29} + 184 q^{31} + 42 q^{33} + 406 q^{37} + 90 q^{39} + 130 q^{41} + 148 q^{43}+ \cdots - 126 q^{99}+O(q^{100}) $

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

−3.00000

22.0000

9.00000

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	300.4.a.d		1
3.b	odd	2	1		900.4.a.o		1
4.b	odd	2	1		1200.4.a.w		1
5.b	even	2	1		300.4.a.f		1
5.c	odd	4	2		60.4.d.a	✓	2
15.d	odd	2	1		900.4.a.d		1
15.e	even	4	2		180.4.d.b		2
20.d	odd	2	1		1200.4.a.q		1
20.e	even	4	2		240.4.f.a		2
40.i	odd	4	2		960.4.f.j		2
40.k	even	4	2		960.4.f.i		2
60.l	odd	4	2		720.4.f.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.4.d.a	✓	2	5.c	odd	4	2
180.4.d.b		2	15.e	even	4	2
240.4.f.a		2	20.e	even	4	2
300.4.a.d		1	1.a	even	1	1	trivial
300.4.a.f		1	5.b	even	2	1
720.4.f.h		2	60.l	odd	4	2
900.4.a.d		1	15.d	odd	2	1
900.4.a.o		1	3.b	odd	2	1
960.4.f.i		2	40.k	even	4	2
960.4.f.j		2	40.i	odd	4	2
1200.4.a.q		1	20.d	odd	2	1
1200.4.a.w		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 22 $ T7 - 22 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(300))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 3 $ T + 3
$5$	$ T $ T
$7$	$ T - 22 $ T - 22
$11$	$ T + 14 $ T + 14
$13$	$ T + 30 $ T + 30
$17$	$ T - 62 $ T - 62
$19$	$ T + 120 $ T + 120
$23$	$ T - 188 $ T - 188
$29$	$ T - 96 $ T - 96
$31$	$ T - 184 $ T - 184
$37$	$ T - 406 $ T - 406
$41$	$ T - 130 $ T - 130
$43$	$ T - 148 $ T - 148
$47$	$ T - 448 $ T - 448
$53$	$ T + 414 $ T + 414
$59$	$ T - 266 $ T - 266
$61$	$ T + 838 $ T + 838
$67$	$ T - 248 $ T - 248
$71$	$ T - 1020 $ T - 1020
$73$	$ T - 484 $ T - 484
$79$	$ T + 48 $ T + 48
$83$	$ T - 548 $ T - 548
$89$	$ T + 650 $ T + 650
$97$	$ T + 1816 $ T + 1816
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	300.a (trivial)

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