LMFDB - Newform orbit 36.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [36,4,Mod(1,36)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, 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("36.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$36 = 2^{2} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	36.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.12406876021$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 12)
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 + 18 q^{5} + 8 q^{7} - 36 q^{11} - 10 q^{13} - 18 q^{17} - 100 q^{19} - 72 q^{23} + 199 q^{25} + 234 q^{29} - 16 q^{31} + 144 q^{35} - 226 q^{37} - 90 q^{41} + 452 q^{43} - 432 q^{47} - 279 q^{49} - 414 q^{53}+ \cdots - 1054 q^{97}+O(q^{100})

q + 18 * q^5 + 8 * q^7 - 36 * q^11 - 10 * q^13 - 18 * q^17 - 100 * q^19 - 72 * q^23 + 199 * q^25 + 234 * q^29 - 16 * q^31 + 144 * q^35 - 226 * q^37 - 90 * q^41 + 452 * q^43 - 432 * q^47 - 279 * q^49 - 414 * q^53 - 648 * q^55 + 684 * q^59 + 422 * q^61 - 180 * q^65 + 332 * q^67 + 360 * q^71 + 26 * q^73 - 288 * q^77 + 512 * q^79 + 1188 * q^83 - 324 * q^85 + 630 * q^89 - 80 * q^91 - 1800 * q^95 - 1054 * q^97

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

18.0000

8.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$

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	36.4.a.a		1
3.b	odd	2	1		12.4.a.a	✓	1
4.b	odd	2	1		144.4.a.g		1
5.b	even	2	1		900.4.a.g		1
5.c	odd	4	2		900.4.d.c		2
7.b	odd	2	1		1764.4.a.b		1
7.c	even	3	2		1764.4.k.b		2
7.d	odd	6	2		1764.4.k.o		2
8.b	even	2	1		576.4.a.b		1
8.d	odd	2	1		576.4.a.a		1
9.c	even	3	2		324.4.e.a		2
9.d	odd	6	2		324.4.e.h		2
12.b	even	2	1		48.4.a.a		1
15.d	odd	2	1		300.4.a.b		1
15.e	even	4	2		300.4.d.e		2
21.c	even	2	1		588.4.a.c		1
21.g	even	6	2		588.4.i.e		2
21.h	odd	6	2		588.4.i.d		2
24.f	even	2	1		192.4.a.l		1
24.h	odd	2	1		192.4.a.f		1
33.d	even	2	1		1452.4.a.d		1
39.d	odd	2	1		2028.4.a.c		1
39.f	even	4	2		2028.4.b.c		2
48.i	odd	4	2		768.4.d.g		2
48.k	even	4	2		768.4.d.j		2
60.h	even	2	1		1200.4.a.be		1
60.l	odd	4	2		1200.4.f.d		2
84.h	odd	2	1		2352.4.a.bk		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.4.a.a	✓	1	3.b	odd	2	1
36.4.a.a		1	1.a	even	1	1	trivial
48.4.a.a		1	12.b	even	2	1
144.4.a.g		1	4.b	odd	2	1
192.4.a.f		1	24.h	odd	2	1
192.4.a.l		1	24.f	even	2	1
300.4.a.b		1	15.d	odd	2	1
300.4.d.e		2	15.e	even	4	2
324.4.e.a		2	9.c	even	3	2
324.4.e.h		2	9.d	odd	6	2
576.4.a.a		1	8.d	odd	2	1
576.4.a.b		1	8.b	even	2	1
588.4.a.c		1	21.c	even	2	1
588.4.i.d		2	21.h	odd	6	2
588.4.i.e		2	21.g	even	6	2
768.4.d.g		2	48.i	odd	4	2
768.4.d.j		2	48.k	even	4	2
900.4.a.g		1	5.b	even	2	1
900.4.d.c		2	5.c	odd	4	2
1200.4.a.be		1	60.h	even	2	1
1200.4.f.d		2	60.l	odd	4	2
1452.4.a.d		1	33.d	even	2	1
1764.4.a.b		1	7.b	odd	2	1
1764.4.k.b		2	7.c	even	3	2
1764.4.k.o		2	7.d	odd	6	2
2028.4.a.c		1	39.d	odd	2	1
2028.4.b.c		2	39.f	even	4	2
2352.4.a.bk		1	84.h	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(\Gamma_0(36))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 18$ T - 18
$7$	$T - 8$ T - 8
$11$	$T + 36$ T + 36
$13$	$T + 10$ T + 10
$17$	$T + 18$ T + 18
$19$	$T + 100$ T + 100
$23$	$T + 72$ T + 72
$29$	$T - 234$ T - 234
$31$	$T + 16$ T + 16
$37$	$T + 226$ T + 226
$41$	$T + 90$ T + 90
$43$	$T - 452$ T - 452
$47$	$T + 432$ T + 432
$53$	$T + 414$ T + 414
$59$	$T - 684$ T - 684
$61$	$T - 422$ T - 422
$67$	$T - 332$ T - 332
$71$	$T - 360$ T - 360
$73$	$T - 26$ T - 26
$79$	$T - 512$ T - 512
$83$	$T - 1188$ T - 1188
$89$	$T - 630$ T - 630
$97$	$T + 1054$ T + 1054
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