LMFDB - Newform orbit 300.4.a.e

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} - 32 q^{7} + 9 q^{9}+O(q^{10}) $

$ q + 3 q^{3} - 32 q^{7} + 9 q^{9} + 36 q^{11} + 10 q^{13} + 78 q^{17} + 140 q^{19} - 96 q^{21} + 192 q^{23} + 27 q^{27} + 6 q^{29} - 16 q^{31} + 108 q^{33} + 34 q^{37} + 30 q^{39} - 390 q^{41} + 52 q^{43} - 408 q^{47} + 681 q^{49} + 234 q^{51} + 114 q^{53} + 420 q^{57} + 516 q^{59} - 58 q^{61} - 288 q^{63} + 892 q^{67} + 576 q^{69} - 120 q^{71} + 646 q^{73} - 1152 q^{77} - 1168 q^{79} + 81 q^{81} + 732 q^{83} + 18 q^{87} - 1590 q^{89} - 320 q^{91} - 48 q^{93} - 194 q^{97} + 324 q^{99}+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

3.00000

−32.0000

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.e		1
3.b	odd	2	1		900.4.a.b		1
4.b	odd	2	1		1200.4.a.s		1
5.b	even	2	1		60.4.a.b	✓	1
5.c	odd	4	2		300.4.d.d		2
15.d	odd	2	1		180.4.a.c		1
15.e	even	4	2		900.4.d.b		2
20.d	odd	2	1		240.4.a.j		1
20.e	even	4	2		1200.4.f.e		2
40.e	odd	2	1		960.4.a.a		1
40.f	even	2	1		960.4.a.bb		1
45.h	odd	6	2		1620.4.i.g		2
45.j	even	6	2		1620.4.i.a		2
60.h	even	2	1		720.4.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.4.a.b	✓	1	5.b	even	2	1
180.4.a.c		1	15.d	odd	2	1
240.4.a.j		1	20.d	odd	2	1
300.4.a.e		1	1.a	even	1	1	trivial
300.4.d.d		2	5.c	odd	4	2
720.4.a.c		1	60.h	even	2	1
900.4.a.b		1	3.b	odd	2	1
900.4.d.b		2	15.e	even	4	2
960.4.a.a		1	40.e	odd	2	1
960.4.a.bb		1	40.f	even	2	1
1200.4.a.s		1	4.b	odd	2	1
1200.4.f.e		2	20.e	even	4	2
1620.4.i.a		2	45.j	even	6	2
1620.4.i.g		2	45.h	odd	6	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 32 $ T7 + 32 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 + 32 $ T + 32
$11$	$ T - 36 $ T - 36
$13$	$ T - 10 $ T - 10
$17$	$ T - 78 $ T - 78
$19$	$ T - 140 $ T - 140
$23$	$ T - 192 $ T - 192
$29$	$ T - 6 $ T - 6
$31$	$ T + 16 $ T + 16
$37$	$ T - 34 $ T - 34
$41$	$ T + 390 $ T + 390
$43$	$ T - 52 $ T - 52
$47$	$ T + 408 $ T + 408
$53$	$ T - 114 $ T - 114
$59$	$ T - 516 $ T - 516
$61$	$ T + 58 $ T + 58
$67$	$ T - 892 $ T - 892
$71$	$ T + 120 $ T + 120
$73$	$ T - 646 $ T - 646
$79$	$ T + 1168 $ T + 1168
$83$	$ T - 732 $ T - 732
$89$	$ T + 1590 $ T + 1590
$97$	$ T + 194 $ T + 194
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Elliptic curves over $\Q$
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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
Significant digits:
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