LMFDB - Newform orbit 126.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,4,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, 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("126.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$126 = 2 \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.43424066072$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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} - 18 q^{5} + 7 q^{7} - 8 q^{8} + 36 q^{10} + 72 q^{11} - 34 q^{13} - 14 q^{14} + 16 q^{16} - 6 q^{17} + 92 q^{19} - 72 q^{20} - 144 q^{22} + 180 q^{23} + 199 q^{25} + 68 q^{26} + 28 q^{28}+ \cdots - 98 q^{98}+O(q^{100})

q - 2 * q^2 + 4 * q^4 - 18 * q^5 + 7 * q^7 - 8 * q^8 + 36 * q^10 + 72 * q^11 - 34 * q^13 - 14 * q^14 + 16 * q^16 - 6 * q^17 + 92 * q^19 - 72 * q^20 - 144 * q^22 + 180 * q^23 + 199 * q^25 + 68 * q^26 + 28 * q^28 + 114 * q^29 + 56 * q^31 - 32 * q^32 + 12 * q^34 - 126 * q^35 - 34 * q^37 - 184 * q^38 + 144 * q^40 - 6 * q^41 + 164 * q^43 + 288 * q^44 - 360 * q^46 - 168 * q^47 + 49 * q^49 - 398 * q^50 - 136 * q^52 - 654 * q^53 - 1296 * q^55 - 56 * q^56 - 228 * q^58 + 492 * q^59 - 250 * q^61 - 112 * q^62 + 64 * q^64 + 612 * q^65 - 124 * q^67 - 24 * q^68 + 252 * q^70 - 36 * q^71 + 1010 * q^73 + 68 * q^74 + 368 * q^76 + 504 * q^77 + 56 * q^79 - 288 * q^80 + 12 * q^82 - 228 * q^83 + 108 * q^85 - 328 * q^86 - 576 * q^88 - 390 * q^89 - 238 * q^91 + 720 * q^92 + 336 * q^94 - 1656 * q^95 - 70 * q^97 - 98 * 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

−18.0000

7.00000

−8.00000

36.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$7$	$-1$

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	126.4.a.a		1
3.b	odd	2	1		42.4.a.a	✓	1
4.b	odd	2	1		1008.4.a.b		1
7.b	odd	2	1		882.4.a.g		1
7.c	even	3	2		882.4.g.w		2
7.d	odd	6	2		882.4.g.o		2
12.b	even	2	1		336.4.a.l		1
15.d	odd	2	1		1050.4.a.g		1
15.e	even	4	2		1050.4.g.a		2
21.c	even	2	1		294.4.a.i		1
21.g	even	6	2		294.4.e.b		2
21.h	odd	6	2		294.4.e.c		2
24.f	even	2	1		1344.4.a.a		1
24.h	odd	2	1		1344.4.a.o		1
84.h	odd	2	1		2352.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.a	✓	1	3.b	odd	2	1
126.4.a.a		1	1.a	even	1	1	trivial
294.4.a.i		1	21.c	even	2	1
294.4.e.b		2	21.g	even	6	2
294.4.e.c		2	21.h	odd	6	2
336.4.a.l		1	12.b	even	2	1
882.4.a.g		1	7.b	odd	2	1
882.4.g.o		2	7.d	odd	6	2
882.4.g.w		2	7.c	even	3	2
1008.4.a.b		1	4.b	odd	2	1
1050.4.a.g		1	15.d	odd	2	1
1050.4.g.a		2	15.e	even	4	2
1344.4.a.a		1	24.f	even	2	1
1344.4.a.o		1	24.h	odd	2	1
2352.4.a.a		1	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 18$ T5 + 18 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(126))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T$ T
$5$	$T + 18$ T + 18
$7$	$T - 7$ T - 7
$11$	$T - 72$ T - 72
$13$	$T + 34$ T + 34
$17$	$T + 6$ T + 6
$19$	$T - 92$ T - 92
$23$	$T - 180$ T - 180
$29$	$T - 114$ T - 114
$31$	$T - 56$ T - 56
$37$	$T + 34$ T + 34
$41$	$T + 6$ T + 6
$43$	$T - 164$ T - 164
$47$	$T + 168$ T + 168
$53$	$T + 654$ T + 654
$59$	$T - 492$ T - 492
$61$	$T + 250$ T + 250
$67$	$T + 124$ T + 124
$71$	$T + 36$ T + 36
$73$	$T - 1010$ T - 1010
$79$	$T - 56$ T - 56
$83$	$T + 228$ T + 228
$89$	$T + 390$ T + 390
$97$	$T + 70$ T + 70
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