LMFDB - Newform orbit 450.4.a.i

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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
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} + 23 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} + 23 q^{7} - 8 q^{8} + 30 q^{11} + 29 q^{13} - 46 q^{14} + 16 q^{16} - 78 q^{17} + 149 q^{19} - 60 q^{22} - 150 q^{23} - 58 q^{26} + 92 q^{28} + 234 q^{29} - 217 q^{31} - 32 q^{32} + 156 q^{34} + 146 q^{37} - 298 q^{38} + 156 q^{41} - 433 q^{43} + 120 q^{44} + 300 q^{46} - 30 q^{47} + 186 q^{49} + 116 q^{52} + 552 q^{53} - 184 q^{56} - 468 q^{58} + 270 q^{59} + 275 q^{61} + 434 q^{62} + 64 q^{64} + 803 q^{67} - 312 q^{68} - 660 q^{71} - 646 q^{73} - 292 q^{74} + 596 q^{76} + 690 q^{77} + 992 q^{79} - 312 q^{82} + 846 q^{83} + 866 q^{86} - 240 q^{88} + 1488 q^{89} + 667 q^{91} - 600 q^{92} + 60 q^{94} - 319 q^{97} - 372 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

23.0000

−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.i		1
3.b	odd	2	1		150.4.a.g	yes	1
5.b	even	2	1		450.4.a.l		1
5.c	odd	4	2		450.4.c.h		2
12.b	even	2	1		1200.4.a.v		1
15.d	odd	2	1		150.4.a.c	✓	1
15.e	even	4	2		150.4.c.b		2
60.h	even	2	1		1200.4.a.r		1
60.l	odd	4	2		1200.4.f.q		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
150.4.a.c	✓	1	15.d	odd	2	1
150.4.a.g	yes	1	3.b	odd	2	1
150.4.c.b		2	15.e	even	4	2
450.4.a.i		1	1.a	even	1	1	trivial
450.4.a.l		1	5.b	even	2	1
450.4.c.h		2	5.c	odd	4	2
1200.4.a.r		1	60.h	even	2	1
1200.4.a.v		1	12.b	even	2	1
1200.4.f.q		2	60.l	odd	4	2

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} - 23 $

$ T_{11} - 30 $

$ T_{17} + 78 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 23 $ T - 23
$11$	$ T - 30 $ T - 30
$13$	$ T - 29 $ T - 29
$17$	$ T + 78 $ T + 78
$19$	$ T - 149 $ T - 149
$23$	$ T + 150 $ T + 150
$29$	$ T - 234 $ T - 234
$31$	$ T + 217 $ T + 217
$37$	$ T - 146 $ T - 146
$41$	$ T - 156 $ T - 156
$43$	$ T + 433 $ T + 433
$47$	$ T + 30 $ T + 30
$53$	$ T - 552 $ T - 552
$59$	$ T - 270 $ T - 270
$61$	$ T - 275 $ T - 275
$67$	$ T - 803 $ T - 803
$71$	$ T + 660 $ T + 660
$73$	$ T + 646 $ T + 646
$79$	$ T - 992 $ T - 992
$83$	$ T - 846 $ T - 846
$89$	$ T - 1488 $ T - 1488
$97$	$ T + 319 $ T + 319
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Classical	Maass
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Elliptic curves over $\Q$
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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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