LMFDB - Newform orbit 1344.4.a.b

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,-14,0,-7,0,9,0,-4] 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 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} - 14 q^{5} - 7 q^{7} + 9 q^{9} - 4 q^{11} - 54 q^{13} + 42 q^{15} - 14 q^{17} - 92 q^{19} + 21 q^{21} - 152 q^{23} + 71 q^{25} - 27 q^{27} + 106 q^{29} - 144 q^{31} + 12 q^{33} + 98 q^{35}+ \cdots - 36 q^{99}+O(q^{100})

q - 3 * q^3 - 14 * q^5 - 7 * q^7 + 9 * q^9 - 4 * q^11 - 54 * q^13 + 42 * q^15 - 14 * q^17 - 92 * q^19 + 21 * q^21 - 152 * q^23 + 71 * q^25 - 27 * q^27 + 106 * q^29 - 144 * q^31 + 12 * q^33 + 98 * q^35 - 158 * q^37 + 162 * q^39 - 390 * q^41 + 508 * q^43 - 126 * q^45 - 528 * q^47 + 49 * q^49 + 42 * q^51 - 606 * q^53 + 56 * q^55 + 276 * q^57 + 364 * q^59 - 678 * q^61 - 63 * q^63 + 756 * q^65 - 844 * q^67 + 456 * q^69 - 8 * q^71 - 422 * q^73 - 213 * q^75 + 28 * q^77 + 384 * q^79 + 81 * q^81 + 548 * q^83 + 196 * q^85 - 318 * q^87 + 1194 * q^89 + 378 * q^91 + 432 * q^93 + 1288 * q^95 - 1502 * q^97 - 36 * 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

−14.0000

−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.b		1
4.b	odd	2	1		1344.4.a.p		1
8.b	even	2	1		84.4.a.b	✓	1
8.d	odd	2	1		336.4.a.e		1
24.f	even	2	1		1008.4.a.d		1
24.h	odd	2	1		252.4.a.a		1
40.f	even	2	1		2100.4.a.g		1
40.i	odd	4	2		2100.4.k.g		2
56.e	even	2	1		2352.4.a.v		1
56.h	odd	2	1		588.4.a.a		1
56.j	odd	6	2		588.4.i.h		2
56.p	even	6	2		588.4.i.a		2
168.i	even	2	1		1764.4.a.l		1
168.s	odd	6	2		1764.4.k.n		2
168.ba	even	6	2		1764.4.k.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.4.a.b	✓	1	8.b	even	2	1
252.4.a.a		1	24.h	odd	2	1
336.4.a.e		1	8.d	odd	2	1
588.4.a.a		1	56.h	odd	2	1
588.4.i.a		2	56.p	even	6	2
588.4.i.h		2	56.j	odd	6	2
1008.4.a.d		1	24.f	even	2	1
1344.4.a.b		1	1.a	even	1	1	trivial
1344.4.a.p		1	4.b	odd	2	1
1764.4.a.l		1	168.i	even	2	1
1764.4.k.c		2	168.ba	even	6	2
1764.4.k.n		2	168.s	odd	6	2
2100.4.a.g		1	40.f	even	2	1
2100.4.k.g		2	40.i	odd	4	2
2352.4.a.v		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} + 14

T5 + 14

T_{11} + 4

T11 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T + 14$ T + 14
$7$	$T + 7$ T + 7
$11$	$T + 4$ T + 4
$13$	$T + 54$ T + 54
$17$	$T + 14$ T + 14
$19$	$T + 92$ T + 92
$23$	$T + 152$ T + 152
$29$	$T - 106$ T - 106
$31$	$T + 144$ T + 144
$37$	$T + 158$ T + 158
$41$	$T + 390$ T + 390
$43$	$T - 508$ T - 508
$47$	$T + 528$ T + 528
$53$	$T + 606$ T + 606
$59$	$T - 364$ T - 364
$61$	$T + 678$ T + 678
$67$	$T + 844$ T + 844
$71$	$T + 8$ T + 8
$73$	$T + 422$ T + 422
$79$	$T - 384$ T - 384
$83$	$T - 548$ T - 548
$89$	$T - 1194$ T - 1194
$97$	$T + 1502$ T + 1502
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