LMFDB - Newform orbit 630.4.a.o

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} - 60 q^{11} + 38 q^{13} + 14 q^{14} + 16 q^{16} - 42 q^{17} - 52 q^{19} - 20 q^{20} - 120 q^{22} - 120 q^{23} + 25 q^{25} + 76 q^{26} + 28 q^{28} + 234 q^{29} - 304 q^{31} + 32 q^{32} - 84 q^{34} - 35 q^{35} - 106 q^{37} - 104 q^{38} - 40 q^{40} + 54 q^{41} - 196 q^{43} - 240 q^{44} - 240 q^{46} - 336 q^{47} + 49 q^{49} + 50 q^{50} + 152 q^{52} - 438 q^{53} + 300 q^{55} + 56 q^{56} + 468 q^{58} + 444 q^{59} + 38 q^{61} - 608 q^{62} + 64 q^{64} - 190 q^{65} - 988 q^{67} - 168 q^{68} - 70 q^{70} + 720 q^{71} + 146 q^{73} - 212 q^{74} - 208 q^{76} - 420 q^{77} - 808 q^{79} - 80 q^{80} + 108 q^{82} - 612 q^{83} + 210 q^{85} - 392 q^{86} - 480 q^{88} - 1146 q^{89} + 266 q^{91} - 480 q^{92} - 672 q^{94} + 260 q^{95} - 70 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.o		1
3.b	odd	2	1		70.4.a.d	✓	1
12.b	even	2	1		560.4.a.g		1
15.d	odd	2	1		350.4.a.o		1
15.e	even	4	2		350.4.c.d		2
21.c	even	2	1		490.4.a.b		1
21.g	even	6	2		490.4.e.q		2
21.h	odd	6	2		490.4.e.k		2
24.f	even	2	1		2240.4.a.y		1
24.h	odd	2	1		2240.4.a.m		1
105.g	even	2	1		2450.4.a.bm		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.d	✓	1	3.b	odd	2	1
350.4.a.o		1	15.d	odd	2	1
350.4.c.d		2	15.e	even	4	2
490.4.a.b		1	21.c	even	2	1
490.4.e.k		2	21.h	odd	6	2
490.4.e.q		2	21.g	even	6	2
560.4.a.g		1	12.b	even	2	1
630.4.a.o		1	1.a	even	1	1	trivial
2240.4.a.m		1	24.h	odd	2	1
2240.4.a.y		1	24.f	even	2	1
2450.4.a.bm		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} + 60 $

$ T_{13} - 38 $

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 + 60 $ T + 60
$13$	$ T - 38 $ T - 38
$17$	$ T + 42 $ T + 42
$19$	$ T + 52 $ T + 52
$23$	$ T + 120 $ T + 120
$29$	$ T - 234 $ T - 234
$31$	$ T + 304 $ T + 304
$37$	$ T + 106 $ T + 106
$41$	$ T - 54 $ T - 54
$43$	$ T + 196 $ T + 196
$47$	$ T + 336 $ T + 336
$53$	$ T + 438 $ T + 438
$59$	$ T - 444 $ T - 444
$61$	$ T - 38 $ T - 38
$67$	$ T + 988 $ T + 988
$71$	$ T - 720 $ T - 720
$73$	$ T - 146 $ T - 146
$79$	$ T + 808 $ T + 808
$83$	$ T + 612 $ T + 612
$89$	$ T + 1146 $ T + 1146
$97$	$ T + 70 $ T + 70
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

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
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