LMFDB - Newform orbit 450.4.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
Analytic rank:	$1$
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 + 2 q^{2} + 4 q^{4} + 2 q^{7} + 8 q^{8}+O(q^{10}) $

$ q + 2 q^{2} + 4 q^{4} + 2 q^{7} + 8 q^{8} - 70 q^{11} - 54 q^{13} + 4 q^{14} + 16 q^{16} - 22 q^{17} + 24 q^{19} - 140 q^{22} - 100 q^{23} - 108 q^{26} + 8 q^{28} - 216 q^{29} + 208 q^{31} + 32 q^{32} - 44 q^{34} + 254 q^{37} + 48 q^{38} + 206 q^{41} - 292 q^{43} - 280 q^{44} - 200 q^{46} - 320 q^{47} - 339 q^{49} - 216 q^{52} - 402 q^{53} + 16 q^{56} - 432 q^{58} + 370 q^{59} - 550 q^{61} + 416 q^{62} + 64 q^{64} - 728 q^{67} - 88 q^{68} + 540 q^{71} - 604 q^{73} + 508 q^{74} + 96 q^{76} - 140 q^{77} + 792 q^{79} + 412 q^{82} + 404 q^{83} - 584 q^{86} - 560 q^{88} + 938 q^{89} - 108 q^{91} - 400 q^{92} - 640 q^{94} - 56 q^{97} - 678 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

2.00000

8.00000

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	450.4.a.p		1
3.b	odd	2	1		150.4.a.d		1
5.b	even	2	1		450.4.a.e		1
5.c	odd	4	2		90.4.c.a		2
12.b	even	2	1		1200.4.a.h		1
15.d	odd	2	1		150.4.a.f		1
15.e	even	4	2		30.4.c.a	✓	2
20.e	even	4	2		720.4.f.c		2
60.h	even	2	1		1200.4.a.bc		1
60.l	odd	4	2		240.4.f.d		2
120.q	odd	4	2		960.4.f.d		2
120.w	even	4	2		960.4.f.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.4.c.a	✓	2	15.e	even	4	2
90.4.c.a		2	5.c	odd	4	2
150.4.a.d		1	3.b	odd	2	1
150.4.a.f		1	15.d	odd	2	1
240.4.f.d		2	60.l	odd	4	2
450.4.a.e		1	5.b	even	2	1
450.4.a.p		1	1.a	even	1	1	trivial
720.4.f.c		2	20.e	even	4	2
960.4.f.c		2	120.w	even	4	2
960.4.f.d		2	120.q	odd	4	2
1200.4.a.h		1	12.b	even	2	1
1200.4.a.bc		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(450))$:

$ T_{7} - 2 $

$ T_{11} + 70 $

$ T_{17} + 22 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 2 $ T - 2
$11$	$ T + 70 $ T + 70
$13$	$ T + 54 $ T + 54
$17$	$ T + 22 $ T + 22
$19$	$ T - 24 $ T - 24
$23$	$ T + 100 $ T + 100
$29$	$ T + 216 $ T + 216
$31$	$ T - 208 $ T - 208
$37$	$ T - 254 $ T - 254
$41$	$ T - 206 $ T - 206
$43$	$ T + 292 $ T + 292
$47$	$ T + 320 $ T + 320
$53$	$ T + 402 $ T + 402
$59$	$ T - 370 $ T - 370
$61$	$ T + 550 $ T + 550
$67$	$ T + 728 $ T + 728
$71$	$ T - 540 $ T - 540
$73$	$ T + 604 $ T + 604
$79$	$ T - 792 $ T - 792
$83$	$ T - 404 $ T - 404
$89$	$ T - 938 $ T - 938
$97$	$ T + 56 $ T + 56
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

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