LMFDB - Newform orbit 336.4.a.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(1,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,6] 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:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 84)
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 + 3 q^{3} + 6 q^{5} - 7 q^{7} + 9 q^{9} - 36 q^{11} + 62 q^{13} + 18 q^{15} + 114 q^{17} + 76 q^{19} - 21 q^{21} + 24 q^{23} - 89 q^{25} + 27 q^{27} + 54 q^{29} + 112 q^{31} - 108 q^{33} - 42 q^{35}+ \cdots - 324 q^{99}+O(q^{100})

q + 3 * q^3 + 6 * q^5 - 7 * q^7 + 9 * q^9 - 36 * q^11 + 62 * q^13 + 18 * q^15 + 114 * q^17 + 76 * q^19 - 21 * q^21 + 24 * q^23 - 89 * q^25 + 27 * q^27 + 54 * q^29 + 112 * q^31 - 108 * q^33 - 42 * q^35 - 178 * q^37 + 186 * q^39 + 378 * q^41 + 172 * q^43 + 54 * q^45 + 192 * q^47 + 49 * q^49 + 342 * q^51 - 402 * q^53 - 216 * q^55 + 228 * q^57 - 396 * q^59 + 254 * q^61 - 63 * q^63 + 372 * q^65 + 1012 * q^67 + 72 * q^69 - 840 * q^71 + 890 * q^73 - 267 * q^75 + 252 * q^77 - 80 * q^79 + 81 * q^81 + 108 * q^83 + 684 * q^85 + 162 * q^87 - 1638 * q^89 - 434 * q^91 + 336 * q^93 + 456 * q^95 + 1010 * q^97 - 324 * q^99

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

3.00000

6.00000

−7.00000

9.00000

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	336.4.a.k		1
3.b	odd	2	1		1008.4.a.h		1
4.b	odd	2	1		84.4.a.a	✓	1
7.b	odd	2	1		2352.4.a.d		1
8.b	even	2	1		1344.4.a.d		1
8.d	odd	2	1		1344.4.a.q		1
12.b	even	2	1		252.4.a.b		1
20.d	odd	2	1		2100.4.a.l		1
20.e	even	4	2		2100.4.k.j		2
28.d	even	2	1		588.4.a.d		1
28.f	even	6	2		588.4.i.c		2
28.g	odd	6	2		588.4.i.f		2
84.h	odd	2	1		1764.4.a.j		1
84.j	odd	6	2		1764.4.k.f		2
84.n	even	6	2		1764.4.k.l		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.a.a	✓	1	4.b	odd	2	1
252.4.a.b		1	12.b	even	2	1
336.4.a.k		1	1.a	even	1	1	trivial
588.4.a.d		1	28.d	even	2	1
588.4.i.c		2	28.f	even	6	2
588.4.i.f		2	28.g	odd	6	2
1008.4.a.h		1	3.b	odd	2	1
1344.4.a.d		1	8.b	even	2	1
1344.4.a.q		1	8.d	odd	2	1
1764.4.a.j		1	84.h	odd	2	1
1764.4.k.f		2	84.j	odd	6	2
1764.4.k.l		2	84.n	even	6	2
2100.4.a.l		1	20.d	odd	2	1
2100.4.k.j		2	20.e	even	4	2
2352.4.a.d		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(336))$ :

T_{5} - 6

T5 - 6

T_{11} + 36

T11 + 36

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T - 6$ T - 6
$7$	$T + 7$ T + 7
$11$	$T + 36$ T + 36
$13$	$T - 62$ T - 62
$17$	$T - 114$ T - 114
$19$	$T - 76$ T - 76
$23$	$T - 24$ T - 24
$29$	$T - 54$ T - 54
$31$	$T - 112$ T - 112
$37$	$T + 178$ T + 178
$41$	$T - 378$ T - 378
$43$	$T - 172$ T - 172
$47$	$T - 192$ T - 192
$53$	$T + 402$ T + 402
$59$	$T + 396$ T + 396
$61$	$T - 254$ T - 254
$67$	$T - 1012$ T - 1012
$71$	$T + 840$ T + 840
$73$	$T - 890$ T - 890
$79$	$T + 80$ T + 80
$83$	$T - 108$ T - 108
$89$	$T + 1638$ T + 1638
$97$	$T - 1010$ T - 1010
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