LMFDB - Newform orbit 450.4.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,4,Mod(1,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [1,-2,0,4,0,0,-14,-8,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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 90)
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} - 14 q^{7} - 8 q^{8} - 6 q^{11} - 68 q^{13} + 28 q^{14} + 16 q^{16} + 78 q^{17} + 44 q^{19} + 12 q^{22} + 120 q^{23} + 136 q^{26} - 56 q^{28} - 126 q^{29} - 244 q^{31} - 32 q^{32}+ \cdots + 294 q^{98}+O(q^{100}) $

q - 2 * q^2 + 4 * q^4 - 14 * q^7 - 8 * q^8 - 6 * q^11 - 68 * q^13 + 28 * q^14 + 16 * q^16 + 78 * q^17 + 44 * q^19 + 12 * q^22 + 120 * q^23 + 136 * q^26 - 56 * q^28 - 126 * q^29 - 244 * q^31 - 32 * q^32 - 156 * q^34 + 304 * q^37 - 88 * q^38 + 480 * q^41 - 104 * q^43 - 24 * q^44 - 240 * q^46 + 600 * q^47 - 147 * q^49 - 272 * q^52 - 258 * q^53 + 112 * q^56 + 252 * q^58 - 534 * q^59 + 362 * q^61 + 488 * q^62 + 64 * q^64 + 268 * q^67 + 312 * q^68 + 972 * q^71 - 470 * q^73 - 608 * q^74 + 176 * q^76 + 84 * q^77 + 1244 * q^79 - 960 * q^82 + 396 * q^83 + 208 * q^86 + 48 * q^88 + 972 * q^89 + 952 * q^91 + 480 * q^92 - 1200 * q^94 + 46 * q^97 + 294 * q^98

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

−14.0000

−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.c		1
3.b	odd	2	1		450.4.a.m		1
5.b	even	2	1		90.4.a.e	yes	1
5.c	odd	4	2		450.4.c.f		2
15.d	odd	2	1		90.4.a.b	✓	1
15.e	even	4	2		450.4.c.g		2
20.d	odd	2	1		720.4.a.t		1
45.h	odd	6	2		810.4.e.u		2
45.j	even	6	2		810.4.e.a		2
60.h	even	2	1		720.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.4.a.b	✓	1	15.d	odd	2	1
90.4.a.e	yes	1	5.b	even	2	1
450.4.a.c		1	1.a	even	1	1	trivial
450.4.a.m		1	3.b	odd	2	1
450.4.c.f		2	5.c	odd	4	2
450.4.c.g		2	15.e	even	4	2
720.4.a.e		1	60.h	even	2	1
720.4.a.t		1	20.d	odd	2	1
810.4.e.a		2	45.j	even	6	2
810.4.e.u		2	45.h	odd	6	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} + 14 $

T7 + 14

$ T_{11} + 6 $

T11 + 6

$ T_{17} - 78 $

T17 - 78

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 14 $ T + 14
$11$	$ T + 6 $ T + 6
$13$	$ T + 68 $ T + 68
$17$	$ T - 78 $ T - 78
$19$	$ T - 44 $ T - 44
$23$	$ T - 120 $ T - 120
$29$	$ T + 126 $ T + 126
$31$	$ T + 244 $ T + 244
$37$	$ T - 304 $ T - 304
$41$	$ T - 480 $ T - 480
$43$	$ T + 104 $ T + 104
$47$	$ T - 600 $ T - 600
$53$	$ T + 258 $ T + 258
$59$	$ T + 534 $ T + 534
$61$	$ T - 362 $ T - 362
$67$	$ T - 268 $ T - 268
$71$	$ T - 972 $ T - 972
$73$	$ T + 470 $ T + 470
$79$	$ T - 1244 $ T - 1244
$83$	$ T - 396 $ T - 396
$89$	$ T - 972 $ T - 972
$97$	$ T - 46 $ T - 46
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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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