LMFDB - Newform orbit 336.4.a.c

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)

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")

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);

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,-2] 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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
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} - 2 q^{5} + 7 q^{7} + 9 q^{9} - 52 q^{11} + 86 q^{13} + 6 q^{15} - 30 q^{17} + 4 q^{19} - 21 q^{21} - 120 q^{23} - 121 q^{25} - 27 q^{27} + 246 q^{29} - 80 q^{31} + 156 q^{33} - 14 q^{35} - 290 q^{37}+ \cdots - 468 q^{99}+O(q^{100})

q - 3 * q^3 - 2 * q^5 + 7 * q^7 + 9 * q^9 - 52 * q^11 + 86 * q^13 + 6 * q^15 - 30 * q^17 + 4 * q^19 - 21 * q^21 - 120 * q^23 - 121 * q^25 - 27 * q^27 + 246 * q^29 - 80 * q^31 + 156 * q^33 - 14 * q^35 - 290 * q^37 - 258 * q^39 - 374 * q^41 - 164 * q^43 - 18 * q^45 - 464 * q^47 + 49 * q^49 + 90 * q^51 - 162 * q^53 + 104 * q^55 - 12 * q^57 - 180 * q^59 - 666 * q^61 + 63 * q^63 - 172 * q^65 + 628 * q^67 + 360 * q^69 - 296 * q^71 - 518 * q^73 + 363 * q^75 - 364 * q^77 + 1184 * q^79 + 81 * q^81 - 220 * q^83 + 60 * q^85 - 738 * q^87 - 774 * q^89 + 602 * q^91 + 240 * q^93 - 8 * q^95 - 1086 * q^97 - 468 * 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

−2.00000

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.c		1
3.b	odd	2	1		1008.4.a.l		1
4.b	odd	2	1		168.4.a.f	✓	1
7.b	odd	2	1		2352.4.a.bb		1
8.b	even	2	1		1344.4.a.v		1
8.d	odd	2	1		1344.4.a.g		1
12.b	even	2	1		504.4.a.c		1
28.d	even	2	1		1176.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.f	✓	1	4.b	odd	2	1
336.4.a.c		1	1.a	even	1	1	trivial
504.4.a.c		1	12.b	even	2	1
1008.4.a.l		1	3.b	odd	2	1
1176.4.a.e		1	28.d	even	2	1
1344.4.a.g		1	8.d	odd	2	1
1344.4.a.v		1	8.b	even	2	1
2352.4.a.bb		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} + 2

T5 + 2

T_{11} + 52

T11 + 52

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T + 2$ T + 2
$7$	$T - 7$ T - 7
$11$	$T + 52$ T + 52
$13$	$T - 86$ T - 86
$17$	$T + 30$ T + 30
$19$	$T - 4$ T - 4
$23$	$T + 120$ T + 120
$29$	$T - 246$ T - 246
$31$	$T + 80$ T + 80
$37$	$T + 290$ T + 290
$41$	$T + 374$ T + 374
$43$	$T + 164$ T + 164
$47$	$T + 464$ T + 464
$53$	$T + 162$ T + 162
$59$	$T + 180$ T + 180
$61$	$T + 666$ T + 666
$67$	$T - 628$ T - 628
$71$	$T + 296$ T + 296
$73$	$T + 518$ T + 518
$79$	$T - 1184$ T - 1184
$83$	$T + 220$ T + 220
$89$	$T + 774$ T + 774
$97$	$T + 1086$ T + 1086
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