LMFDB - Newform orbit 168.4.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [168,4,Mod(1,168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, 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("168.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$168 = 2^{3} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	168.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,-10] 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:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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} - 10 q^{5} + 7 q^{7} + 9 q^{9} - 12 q^{11} + 30 q^{13} + 30 q^{15} + 34 q^{17} + 148 q^{19} - 21 q^{21} + 152 q^{23} - 25 q^{25} - 27 q^{27} - 106 q^{29} + 304 q^{31} + 36 q^{33} - 70 q^{35}+ \cdots - 108 q^{99}+O(q^{100})

q - 3 * q^3 - 10 * q^5 + 7 * q^7 + 9 * q^9 - 12 * q^11 + 30 * q^13 + 30 * q^15 + 34 * q^17 + 148 * q^19 - 21 * q^21 + 152 * q^23 - 25 * q^25 - 27 * q^27 - 106 * q^29 + 304 * q^31 + 36 * q^33 - 70 * q^35 - 114 * q^37 - 90 * q^39 + 202 * q^41 + 116 * q^43 - 90 * q^45 + 224 * q^47 + 49 * q^49 - 102 * q^51 - 274 * q^53 + 120 * q^55 - 444 * q^57 - 660 * q^59 + 382 * q^61 + 63 * q^63 - 300 * q^65 + 12 * q^67 - 456 * q^69 - 552 * q^71 - 614 * q^73 + 75 * q^75 - 84 * q^77 + 880 * q^79 + 81 * q^81 - 108 * q^83 - 340 * q^85 + 318 * q^87 - 86 * q^89 + 210 * q^91 - 912 * q^93 - 1480 * q^95 + 1426 * q^97 - 108 * 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

−10.0000

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	168.4.a.a	✓	1
3.b	odd	2	1		504.4.a.f		1
4.b	odd	2	1		336.4.a.g		1
7.b	odd	2	1		1176.4.a.m		1
8.b	even	2	1		1344.4.a.y		1
8.d	odd	2	1		1344.4.a.j		1
12.b	even	2	1		1008.4.a.p		1
28.d	even	2	1		2352.4.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.a	✓	1	1.a	even	1	1	trivial
336.4.a.g		1	4.b	odd	2	1
504.4.a.f		1	3.b	odd	2	1
1008.4.a.p		1	12.b	even	2	1
1176.4.a.m		1	7.b	odd	2	1
1344.4.a.j		1	8.d	odd	2	1
1344.4.a.y		1	8.b	even	2	1
2352.4.a.n		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(168))$ :

T_{5} + 10

T5 + 10

T_{11} + 12

T11 + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 3$ T + 3
$5$	$T + 10$ T + 10
$7$	$T - 7$ T - 7
$11$	$T + 12$ T + 12
$13$	$T - 30$ T - 30
$17$	$T - 34$ T - 34
$19$	$T - 148$ T - 148
$23$	$T - 152$ T - 152
$29$	$T + 106$ T + 106
$31$	$T - 304$ T - 304
$37$	$T + 114$ T + 114
$41$	$T - 202$ T - 202
$43$	$T - 116$ T - 116
$47$	$T - 224$ T - 224
$53$	$T + 274$ T + 274
$59$	$T + 660$ T + 660
$61$	$T - 382$ T - 382
$67$	$T - 12$ T - 12
$71$	$T + 552$ T + 552
$73$	$T + 614$ T + 614
$79$	$T - 880$ T - 880
$83$	$T + 108$ T + 108
$89$	$T + 86$ T + 86
$97$	$T - 1426$ T - 1426
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