LMFDB - Newform orbit 450.4.a.h

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 6)
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} + 16 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} + 16 q^{7} - 8 q^{8} - 12 q^{11} - 38 q^{13} - 32 q^{14} + 16 q^{16} - 126 q^{17} + 20 q^{19} + 24 q^{22} + 168 q^{23} + 76 q^{26} + 64 q^{28} - 30 q^{29} - 88 q^{31} - 32 q^{32} + 252 q^{34} - 254 q^{37} - 40 q^{38} - 42 q^{41} + 52 q^{43} - 48 q^{44} - 336 q^{46} - 96 q^{47} - 87 q^{49} - 152 q^{52} + 198 q^{53} - 128 q^{56} + 60 q^{58} + 660 q^{59} - 538 q^{61} + 176 q^{62} + 64 q^{64} - 884 q^{67} - 504 q^{68} - 792 q^{71} - 218 q^{73} + 508 q^{74} + 80 q^{76} - 192 q^{77} - 520 q^{79} + 84 q^{82} - 492 q^{83} - 104 q^{86} + 96 q^{88} - 810 q^{89} - 608 q^{91} + 672 q^{92} + 192 q^{94} - 1154 q^{97} + 174 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

16.0000

−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.h		1
3.b	odd	2	1		150.4.a.i		1
5.b	even	2	1		18.4.a.a		1
5.c	odd	4	2		450.4.c.e		2
12.b	even	2	1		1200.4.a.b		1
15.d	odd	2	1		6.4.a.a	✓	1
15.e	even	4	2		150.4.c.d		2
20.d	odd	2	1		144.4.a.c		1
35.c	odd	2	1		882.4.a.n		1
35.i	odd	6	2		882.4.g.f		2
35.j	even	6	2		882.4.g.i		2
40.e	odd	2	1		576.4.a.r		1
40.f	even	2	1		576.4.a.q		1
45.h	odd	6	2		162.4.c.f		2
45.j	even	6	2		162.4.c.c		2
55.d	odd	2	1		2178.4.a.e		1
60.h	even	2	1		48.4.a.c		1
60.l	odd	4	2		1200.4.f.j		2
105.g	even	2	1		294.4.a.e		1
105.o	odd	6	2		294.4.e.h		2
105.p	even	6	2		294.4.e.g		2
120.i	odd	2	1		192.4.a.i		1
120.m	even	2	1		192.4.a.c		1
165.d	even	2	1		726.4.a.f		1
195.e	odd	2	1		1014.4.a.g		1
195.n	even	4	2		1014.4.b.d		2
240.t	even	4	2		768.4.d.c		2
240.bm	odd	4	2		768.4.d.n		2
255.h	odd	2	1		1734.4.a.d		1
285.b	even	2	1		2166.4.a.i		1
420.o	odd	2	1		2352.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.4.a.a	✓	1	15.d	odd	2	1
18.4.a.a		1	5.b	even	2	1
48.4.a.c		1	60.h	even	2	1
144.4.a.c		1	20.d	odd	2	1
150.4.a.i		1	3.b	odd	2	1
150.4.c.d		2	15.e	even	4	2
162.4.c.c		2	45.j	even	6	2
162.4.c.f		2	45.h	odd	6	2
192.4.a.c		1	120.m	even	2	1
192.4.a.i		1	120.i	odd	2	1
294.4.a.e		1	105.g	even	2	1
294.4.e.g		2	105.p	even	6	2
294.4.e.h		2	105.o	odd	6	2
450.4.a.h		1	1.a	even	1	1	trivial
450.4.c.e		2	5.c	odd	4	2
576.4.a.q		1	40.f	even	2	1
576.4.a.r		1	40.e	odd	2	1
726.4.a.f		1	165.d	even	2	1
768.4.d.c		2	240.t	even	4	2
768.4.d.n		2	240.bm	odd	4	2
882.4.a.n		1	35.c	odd	2	1
882.4.g.f		2	35.i	odd	6	2
882.4.g.i		2	35.j	even	6	2
1014.4.a.g		1	195.e	odd	2	1
1014.4.b.d		2	195.n	even	4	2
1200.4.a.b		1	12.b	even	2	1
1200.4.f.j		2	60.l	odd	4	2
1734.4.a.d		1	255.h	odd	2	1
2166.4.a.i		1	285.b	even	2	1
2178.4.a.e		1	55.d	odd	2	1
2352.4.a.e		1	420.o	odd	2	1

Hecke kernels

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

$ T_{7} - 16 $

$ T_{11} + 12 $

$ T_{17} + 126 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 16 $ T - 16
$11$	$ T + 12 $ T + 12
$13$	$ T + 38 $ T + 38
$17$	$ T + 126 $ T + 126
$19$	$ T - 20 $ T - 20
$23$	$ T - 168 $ T - 168
$29$	$ T + 30 $ T + 30
$31$	$ T + 88 $ T + 88
$37$	$ T + 254 $ T + 254
$41$	$ T + 42 $ T + 42
$43$	$ T - 52 $ T - 52
$47$	$ T + 96 $ T + 96
$53$	$ T - 198 $ T - 198
$59$	$ T - 660 $ T - 660
$61$	$ T + 538 $ T + 538
$67$	$ T + 884 $ T + 884
$71$	$ T + 792 $ T + 792
$73$	$ T + 218 $ T + 218
$79$	$ T + 520 $ T + 520
$83$	$ T + 492 $ T + 492
$89$	$ T + 810 $ T + 810
$97$	$ T + 1154 $ T + 1154
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Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

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