LMFDB - Newform orbit 7350.2.a.ct

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7350,2,Mod(1,7350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7350, 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("7350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7350.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:	$58.6900454856$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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 + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9}+O(q^{10}) $

$ q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + q^{24} + 2 q^{26} + q^{27} - 6 q^{29} - 8 q^{31} + q^{32} + 6 q^{34} + q^{36} - 2 q^{37} + 4 q^{38} + 2 q^{39} + 6 q^{41} + 4 q^{43} + q^{48} + 6 q^{51} + 2 q^{52} + 6 q^{53} + q^{54} + 4 q^{57} - 6 q^{58} + 10 q^{61} - 8 q^{62} + q^{64} + 4 q^{67} + 6 q^{68} + q^{72} + 2 q^{73} - 2 q^{74} + 4 q^{76} + 2 q^{78} + 8 q^{79} + q^{81} + 6 q^{82} + 12 q^{83} + 4 q^{86} - 6 q^{87} - 18 q^{89} - 8 q^{93} + q^{96} + 2 q^{97}+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

1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$5$	$1$
$7$	$-1$

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	7350.2.a.ct		1
5.b	even	2	1		1470.2.a.d		1
7.b	odd	2	1		150.2.a.b		1
15.d	odd	2	1		4410.2.a.z		1
21.c	even	2	1		450.2.a.d		1
28.d	even	2	1		1200.2.a.k		1
35.c	odd	2	1		30.2.a.a	✓	1
35.f	even	4	2		150.2.c.a		2
35.i	odd	6	2		1470.2.i.o		2
35.j	even	6	2		1470.2.i.q		2
56.e	even	2	1		4800.2.a.d		1
56.h	odd	2	1		4800.2.a.cq		1
84.h	odd	2	1		3600.2.a.f		1
105.g	even	2	1		90.2.a.c		1
105.k	odd	4	2		450.2.c.b		2
140.c	even	2	1		240.2.a.b		1
140.j	odd	4	2		1200.2.f.e		2
280.c	odd	2	1		960.2.a.e		1
280.n	even	2	1		960.2.a.p		1
280.s	even	4	2		4800.2.f.p		2
280.y	odd	4	2		4800.2.f.w		2
315.z	even	6	2		810.2.e.b		2
315.bg	odd	6	2		810.2.e.l		2
385.h	even	2	1		3630.2.a.w		1
420.o	odd	2	1		720.2.a.j		1
420.w	even	4	2		3600.2.f.i		2
455.h	odd	2	1		5070.2.a.w		1
455.u	even	4	2		5070.2.b.k		2
560.be	even	4	2		3840.2.k.f		2
560.bf	odd	4	2		3840.2.k.y		2
595.b	odd	2	1		8670.2.a.g		1
840.b	odd	2	1		2880.2.a.q		1
840.u	even	2	1		2880.2.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.2.a.a	✓	1	35.c	odd	2	1
90.2.a.c		1	105.g	even	2	1
150.2.a.b		1	7.b	odd	2	1
150.2.c.a		2	35.f	even	4	2
240.2.a.b		1	140.c	even	2	1
450.2.a.d		1	21.c	even	2	1
450.2.c.b		2	105.k	odd	4	2
720.2.a.j		1	420.o	odd	2	1
810.2.e.b		2	315.z	even	6	2
810.2.e.l		2	315.bg	odd	6	2
960.2.a.e		1	280.c	odd	2	1
960.2.a.p		1	280.n	even	2	1
1200.2.a.k		1	28.d	even	2	1
1200.2.f.e		2	140.j	odd	4	2
1470.2.a.d		1	5.b	even	2	1
1470.2.i.o		2	35.i	odd	6	2
1470.2.i.q		2	35.j	even	6	2
2880.2.a.a		1	840.u	even	2	1
2880.2.a.q		1	840.b	odd	2	1
3600.2.a.f		1	84.h	odd	2	1
3600.2.f.i		2	420.w	even	4	2
3630.2.a.w		1	385.h	even	2	1
3840.2.k.f		2	560.be	even	4	2
3840.2.k.y		2	560.bf	odd	4	2
4410.2.a.z		1	15.d	odd	2	1
4800.2.a.d		1	56.e	even	2	1
4800.2.a.cq		1	56.h	odd	2	1
4800.2.f.p		2	280.s	even	4	2
4800.2.f.w		2	280.y	odd	4	2
5070.2.a.w		1	455.h	odd	2	1
5070.2.b.k		2	455.u	even	4	2
7350.2.a.ct		1	1.a	even	1	1	trivial
8670.2.a.g		1	595.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7350))$:

$ T_{11} $

$ T_{13} - 2 $

$ T_{17} - 6 $

$ T_{19} - 4 $

$ T_{23} $

$ T_{31} + 8 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T - 2 $ T - 2
$17$	$ T - 6 $ T - 6
$19$	$ T - 4 $ T - 4
$23$	$ T $ T
$29$	$ T + 6 $ T + 6
$31$	$ T + 8 $ T + 8
$37$	$ T + 2 $ T + 2
$41$	$ T - 6 $ T - 6
$43$	$ T - 4 $ T - 4
$47$	$ T $ T
$53$	$ T - 6 $ T - 6
$59$	$ T $ T
$61$	$ T - 10 $ T - 10
$67$	$ T - 4 $ T - 4
$71$	$ T $ T
$73$	$ T - 2 $ T - 2
$79$	$ T - 8 $ T - 8
$83$	$ T - 12 $ T - 12
$89$	$ T + 18 $ T + 18
$97$	$ T - 2 $ T - 2
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7350.a (trivial)

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