LMFDB - Newform orbit 450.8.a.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,8,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, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
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:	$140.573261468$
Analytic rank:	$0$
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 + 8 q^{2} + 64 q^{4} + 1576 q^{7} + 512 q^{8}+O(q^{10}) $

$ q + 8 q^{2} + 64 q^{4} + 1576 q^{7} + 512 q^{8} - 7332 q^{11} + 3802 q^{13} + 12608 q^{14} + 4096 q^{16} - 6606 q^{17} + 24860 q^{19} - 58656 q^{22} + 41448 q^{23} + 30416 q^{26} + 100864 q^{28} + 41610 q^{29} + 33152 q^{31} + 32768 q^{32} - 52848 q^{34} + 36466 q^{37} + 198880 q^{38} + 639078 q^{41} + 156412 q^{43} - 469248 q^{44} + 331584 q^{46} - 433776 q^{47} + 1660233 q^{49} + 243328 q^{52} + 786078 q^{53} + 806912 q^{56} + 332880 q^{58} - 745140 q^{59} - 1660618 q^{61} + 265216 q^{62} + 262144 q^{64} + 3290836 q^{67} - 422784 q^{68} - 5716152 q^{71} - 2659898 q^{73} + 291728 q^{74} + 1591040 q^{76} - 11555232 q^{77} + 3807440 q^{79} + 5112624 q^{82} + 2229468 q^{83} + 1251296 q^{86} - 3753984 q^{88} - 5991210 q^{89} + 5991952 q^{91} + 2652672 q^{92} - 3470208 q^{94} + 4060126 q^{97} + 13281864 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

8.00000

64.0000

1576.00

512.000

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	450.8.a.ba		1
3.b	odd	2	1		150.8.a.e		1
5.b	even	2	1		18.8.a.a		1
5.c	odd	4	2		450.8.c.a		2
15.d	odd	2	1		6.8.a.a	✓	1
15.e	even	4	2		150.8.c.k		2
20.d	odd	2	1		144.8.a.h		1
40.e	odd	2	1		576.8.a.i		1
40.f	even	2	1		576.8.a.h		1
45.h	odd	6	2		162.8.c.d		2
45.j	even	6	2		162.8.c.i		2
60.h	even	2	1		48.8.a.b		1
105.g	even	2	1		294.8.a.l		1
105.o	odd	6	2		294.8.e.c		2
105.p	even	6	2		294.8.e.d		2
120.i	odd	2	1		192.8.a.f		1
120.m	even	2	1		192.8.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.8.a.a	✓	1	15.d	odd	2	1
18.8.a.a		1	5.b	even	2	1
48.8.a.b		1	60.h	even	2	1
144.8.a.h		1	20.d	odd	2	1
150.8.a.e		1	3.b	odd	2	1
150.8.c.k		2	15.e	even	4	2
162.8.c.d		2	45.h	odd	6	2
162.8.c.i		2	45.j	even	6	2
192.8.a.f		1	120.i	odd	2	1
192.8.a.n		1	120.m	even	2	1
294.8.a.l		1	105.g	even	2	1
294.8.e.c		2	105.o	odd	6	2
294.8.e.d		2	105.p	even	6	2
450.8.a.ba		1	1.a	even	1	1	trivial
450.8.c.a		2	5.c	odd	4	2
576.8.a.h		1	40.f	even	2	1
576.8.a.i		1	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{8}^{\mathrm{new}}(\Gamma_0(450))$:

$ T_{7} - 1576 $

$ T_{11} + 7332 $

$ T_{17} + 6606 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 8 $ T - 8
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 1576 $ T - 1576
$11$	$ T + 7332 $ T + 7332
$13$	$ T - 3802 $ T - 3802
$17$	$ T + 6606 $ T + 6606
$19$	$ T - 24860 $ T - 24860
$23$	$ T - 41448 $ T - 41448
$29$	$ T - 41610 $ T - 41610
$31$	$ T - 33152 $ T - 33152
$37$	$ T - 36466 $ T - 36466
$41$	$ T - 639078 $ T - 639078
$43$	$ T - 156412 $ T - 156412
$47$	$ T + 433776 $ T + 433776
$53$	$ T - 786078 $ T - 786078
$59$	$ T + 745140 $ T + 745140
$61$	$ T + 1660618 $ T + 1660618
$67$	$ T - 3290836 $ T - 3290836
$71$	$ T + 5716152 $ T + 5716152
$73$	$ T + 2659898 $ T + 2659898
$79$	$ T - 3807440 $ T - 3807440
$83$	$ T - 2229468 $ T - 2229468
$89$	$ T + 5991210 $ T + 5991210
$97$	$ T - 4060126 $ T - 4060126
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

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