LMFDB - Newform orbit 630.4.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,4,Mod(1,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, 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("630.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	630.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:	$37.1712033036$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
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} + 5 q^{5} + 7 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} + 5 q^{5} + 7 q^{7} - 8 q^{8} - 10 q^{10} + 33 q^{11} - 43 q^{13} - 14 q^{14} + 16 q^{16} - 111 q^{17} - 70 q^{19} + 20 q^{20} - 66 q^{22} - 42 q^{23} + 25 q^{25} + 86 q^{26} + 28 q^{28} + 225 q^{29} - 88 q^{31} - 32 q^{32} + 222 q^{34} + 35 q^{35} - 34 q^{37} + 140 q^{38} - 40 q^{40} - 432 q^{41} - 178 q^{43} + 132 q^{44} + 84 q^{46} - 411 q^{47} + 49 q^{49} - 50 q^{50} - 172 q^{52} + 708 q^{53} + 165 q^{55} - 56 q^{56} - 450 q^{58} - 480 q^{59} + 812 q^{61} + 176 q^{62} + 64 q^{64} - 215 q^{65} + 596 q^{67} - 444 q^{68} - 70 q^{70} - 432 q^{71} - 358 q^{73} + 68 q^{74} - 280 q^{76} + 231 q^{77} + 425 q^{79} + 80 q^{80} + 864 q^{82} - 972 q^{83} - 555 q^{85} + 356 q^{86} - 264 q^{88} - 960 q^{89} - 301 q^{91} - 168 q^{92} + 822 q^{94} - 350 q^{95} - 709 q^{97} - 98 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

5.00000

7.00000

−8.00000

−10.0000

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	630.4.a.j		1
3.b	odd	2	1		70.4.a.f	✓	1
12.b	even	2	1		560.4.a.c		1
15.d	odd	2	1		350.4.a.b		1
15.e	even	4	2		350.4.c.l		2
21.c	even	2	1		490.4.a.i		1
21.g	even	6	2		490.4.e.h		2
21.h	odd	6	2		490.4.e.b		2
24.f	even	2	1		2240.4.a.bh		1
24.h	odd	2	1		2240.4.a.f		1
105.g	even	2	1		2450.4.a.s		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.f	✓	1	3.b	odd	2	1
350.4.a.b		1	15.d	odd	2	1
350.4.c.l		2	15.e	even	4	2
490.4.a.i		1	21.c	even	2	1
490.4.e.b		2	21.h	odd	6	2
490.4.e.h		2	21.g	even	6	2
560.4.a.c		1	12.b	even	2	1
630.4.a.j		1	1.a	even	1	1	trivial
2240.4.a.f		1	24.h	odd	2	1
2240.4.a.bh		1	24.f	even	2	1
2450.4.a.s		1	105.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(630))$:

$ T_{11} - 33 $

$ T_{13} + 43 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T - 5 $ T - 5
$7$	$ T - 7 $ T - 7
$11$	$ T - 33 $ T - 33
$13$	$ T + 43 $ T + 43
$17$	$ T + 111 $ T + 111
$19$	$ T + 70 $ T + 70
$23$	$ T + 42 $ T + 42
$29$	$ T - 225 $ T - 225
$31$	$ T + 88 $ T + 88
$37$	$ T + 34 $ T + 34
$41$	$ T + 432 $ T + 432
$43$	$ T + 178 $ T + 178
$47$	$ T + 411 $ T + 411
$53$	$ T - 708 $ T - 708
$59$	$ T + 480 $ T + 480
$61$	$ T - 812 $ T - 812
$67$	$ T - 596 $ T - 596
$71$	$ T + 432 $ T + 432
$73$	$ T + 358 $ T + 358
$79$	$ T - 425 $ T - 425
$83$	$ T + 972 $ T + 972
$89$	$ T + 960 $ T + 960
$97$	$ T + 709 $ T + 709
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)

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