LMFDB - Newform orbit 3675.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
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} + q^{3} + 2 q^{4} + 2 q^{6} + q^{9}+O(q^{10}) $

$ q + 2 q^{2} + q^{3} + 2 q^{4} + 2 q^{6} + q^{9} - 6 q^{11} + 2 q^{12} - 3 q^{13} - 4 q^{16} - 4 q^{17} + 2 q^{18} - q^{19} - 12 q^{22} + 4 q^{23} - 6 q^{26} + q^{27} - 8 q^{29} - q^{31} - 8 q^{32} - 6 q^{33} - 8 q^{34} + 2 q^{36} - 7 q^{37} - 2 q^{38} - 3 q^{39} + 6 q^{41} - q^{43} - 12 q^{44} + 8 q^{46} + 2 q^{47} - 4 q^{48} - 4 q^{51} - 6 q^{52} - 4 q^{53} + 2 q^{54} - q^{57} - 16 q^{58} + 8 q^{59} + 14 q^{61} - 2 q^{62} - 8 q^{64} - 12 q^{66} - 7 q^{67} - 8 q^{68} + 4 q^{69} + 6 q^{71} + q^{73} - 14 q^{74} - 2 q^{76} - 6 q^{78} - q^{79} + q^{81} + 12 q^{82} + 2 q^{83} - 2 q^{86} - 8 q^{87} + 12 q^{89} + 8 q^{92} - q^{93} + 4 q^{94} - 8 q^{96} - 6 q^{97} - 6 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

2.00000

1.00000

2.00000

1.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$5$	$1$
$7$	$-1$

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	3675.2.a.p		1
5.b	even	2	1		735.2.a.a		1
7.b	odd	2	1		3675.2.a.o		1
7.d	odd	6	2		525.2.i.a		2
15.d	odd	2	1		2205.2.a.m		1
35.c	odd	2	1		735.2.a.b		1
35.i	odd	6	2		105.2.i.b	✓	2
35.j	even	6	2		735.2.i.f		2
35.k	even	12	4		525.2.r.d		4
105.g	even	2	1		2205.2.a.k		1
105.p	even	6	2		315.2.j.a		2
140.s	even	6	2		1680.2.bg.l		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.i.b	✓	2	35.i	odd	6	2
315.2.j.a		2	105.p	even	6	2
525.2.i.a		2	7.d	odd	6	2
525.2.r.d		4	35.k	even	12	4
735.2.a.a		1	5.b	even	2	1
735.2.a.b		1	35.c	odd	2	1
735.2.i.f		2	35.j	even	6	2
1680.2.bg.l		2	140.s	even	6	2
2205.2.a.k		1	105.g	even	2	1
2205.2.a.m		1	15.d	odd	2	1
3675.2.a.o		1	7.b	odd	2	1
3675.2.a.p		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3675))$:

$ T_{2} - 2 $

$ T_{11} + 6 $

$ T_{13} + 3 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 6 $ T + 6
$13$	$ T + 3 $ T + 3
$17$	$ T + 4 $ T + 4
$19$	$ T + 1 $ T + 1
$23$	$ T - 4 $ T - 4
$29$	$ T + 8 $ T + 8
$31$	$ T + 1 $ T + 1
$37$	$ T + 7 $ T + 7
$41$	$ T - 6 $ T - 6
$43$	$ T + 1 $ T + 1
$47$	$ T - 2 $ T - 2
$53$	$ T + 4 $ T + 4
$59$	$ T - 8 $ T - 8
$61$	$ T - 14 $ T - 14
$67$	$ T + 7 $ T + 7
$71$	$ T - 6 $ T - 6
$73$	$ T - 1 $ T - 1
$79$	$ T + 1 $ T + 1
$83$	$ T - 2 $ T - 2
$89$	$ T - 12 $ T - 12
$97$	$ T + 6 $ T + 6
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

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