LMFDB - Newform orbit 1344.4.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(1,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, 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("1344.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1344 = 2^{6} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,-2,0,-7,0,9,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.2985670477$
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 - 3 q^{3} - 2 q^{5} - 7 q^{7} + 9 q^{9} + 8 q^{11} + 42 q^{13} + 6 q^{15} - 2 q^{17} + 124 q^{19} + 21 q^{21} + 76 q^{23} - 121 q^{25} - 27 q^{27} - 254 q^{29} - 72 q^{31} - 24 q^{33} + 14 q^{35} - 398 q^{37}+ \cdots + 72 q^{99}+O(q^{100})

q - 3 * q^3 - 2 * q^5 - 7 * q^7 + 9 * q^9 + 8 * q^11 + 42 * q^13 + 6 * q^15 - 2 * q^17 + 124 * q^19 + 21 * q^21 + 76 * q^23 - 121 * q^25 - 27 * q^27 - 254 * q^29 - 72 * q^31 - 24 * q^33 + 14 * q^35 - 398 * q^37 - 126 * q^39 + 462 * q^41 - 212 * q^43 - 18 * q^45 - 264 * q^47 + 49 * q^49 + 6 * q^51 + 162 * q^53 - 16 * q^55 - 372 * q^57 + 772 * q^59 - 30 * q^61 - 63 * q^63 - 84 * q^65 + 764 * q^67 - 228 * q^69 - 236 * q^71 + 418 * q^73 + 363 * q^75 - 56 * q^77 + 552 * q^79 + 81 * q^81 - 1036 * q^83 + 4 * q^85 + 762 * q^87 + 30 * q^89 - 294 * q^91 + 216 * q^93 - 248 * q^95 - 1190 * q^97 + 72 * 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	1344.4.a.f		1
4.b	odd	2	1		1344.4.a.t		1
8.b	even	2	1		42.4.a.b	✓	1
8.d	odd	2	1		336.4.a.d		1
24.f	even	2	1		1008.4.a.j		1
24.h	odd	2	1		126.4.a.c		1
40.f	even	2	1		1050.4.a.d		1
40.i	odd	4	2		1050.4.g.n		2
56.e	even	2	1		2352.4.a.ba		1
56.h	odd	2	1		294.4.a.h		1
56.j	odd	6	2		294.4.e.d		2
56.p	even	6	2		294.4.e.a		2
168.i	even	2	1		882.4.a.d		1
168.s	odd	6	2		882.4.g.s		2
168.ba	even	6	2		882.4.g.r		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.b	✓	1	8.b	even	2	1
126.4.a.c		1	24.h	odd	2	1
294.4.a.h		1	56.h	odd	2	1
294.4.e.a		2	56.p	even	6	2
294.4.e.d		2	56.j	odd	6	2
336.4.a.d		1	8.d	odd	2	1
882.4.a.d		1	168.i	even	2	1
882.4.g.r		2	168.ba	even	6	2
882.4.g.s		2	168.s	odd	6	2
1008.4.a.j		1	24.f	even	2	1
1050.4.a.d		1	40.f	even	2	1
1050.4.g.n		2	40.i	odd	4	2
1344.4.a.f		1	1.a	even	1	1	trivial
1344.4.a.t		1	4.b	odd	2	1
2352.4.a.ba		1	56.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1344))$ :

T_{5} + 2

T5 + 2

T_{11} - 8

T11 - 8

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
$11$	$T - 8$ T - 8
$13$	$T - 42$ T - 42
$17$	$T + 2$ T + 2
$19$	$T - 124$ T - 124
$23$	$T - 76$ T - 76
$29$	$T + 254$ T + 254
$31$	$T + 72$ T + 72
$37$	$T + 398$ T + 398
$41$	$T - 462$ T - 462
$43$	$T + 212$ T + 212
$47$	$T + 264$ T + 264
$53$	$T - 162$ T - 162
$59$	$T - 772$ T - 772
$61$	$T + 30$ T + 30
$67$	$T - 764$ T - 764
$71$	$T + 236$ T + 236
$73$	$T - 418$ T - 418
$79$	$T - 552$ T - 552
$83$	$T + 1036$ T + 1036
$89$	$T - 30$ T - 30
$97$	$T + 1190$ T + 1190
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