LMFDB - Newform orbit 32.6.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,6,Mod(1,32)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$32 = 2^{5}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$5.13228223402$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{4}$
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)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = 16\sqrt{3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{3} + 46 q^{5} - 6 \beta q^{7} + 525 q^{9} + 3 \beta q^{11} - 42 q^{13} + 46 \beta q^{15} + 962 q^{17} - 75 \beta q^{19} - 4608 q^{21} - 114 \beta q^{23} - 1009 q^{25} + 282 \beta q^{27} - 2554 q^{29} + \cdots + 1575 \beta q^{99} +O(q^{100})$ q + b * q^3 + 46 * q^5 - 6b q^7 + 525 * q^9 + 3b q^11 - 42 * q^13 + 46b q^15 + 962 * q^17 - 75b q^19 - 4608 * q^21 - 114b q^23 - 1009 * q^25 + 282b q^27 - 2554 * q^29 + 72b q^31 + 2304 * q^33 - 276b q^35 + 11950 * q^37 - 42b q^39 - 5078 * q^41 - 453b q^43 + 24150 * q^45 + 444b q^47 + 10841 * q^49 + 962b q^51 - 19714 * q^53 + 138b q^55 - 57600 * q^57 - 321b q^59 + 29318 * q^61 - 3150b q^63 - 1932 * q^65 + 609b q^67 - 87552 * q^69 + 2922b q^71 + 37914 * q^73 - 1009b q^75 - 13824 * q^77 + 3204b q^79 + 89001 * q^81 - 1419b q^83 + 44252 * q^85 - 2554b q^87 + 13930 * q^89 + 252b q^91 + 55296 * q^93 - 3450b q^95 + 163602 * q^97 + 1575b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 92 q^{5} + 1050 q^{9} - 84 q^{13} + 1924 q^{17} - 9216 q^{21} - 2018 q^{25} - 5108 q^{29} + 4608 q^{33} + 23900 q^{37} - 10156 q^{41} + 48300 q^{45} + 21682 q^{49} - 39428 q^{53} - 115200 q^{57} + 58636 q^{61}+ \cdots + 327204 q^{97}+O(q^{100})$ 2 * q + 92 * q^5 + 1050 * q^9 - 84 * q^13 + 1924 * q^17 - 9216 * q^21 - 2018 * q^25 - 5108 * q^29 + 4608 * q^33 + 23900 * q^37 - 10156 * q^41 + 48300 * q^45 + 21682 * q^49 - 39428 * q^53 - 115200 * q^57 + 58636 * q^61 - 3864 * q^65 - 175104 * q^69 + 75828 * q^73 - 27648 * q^77 + 178002 * q^81 + 88504 * q^85 + 27860 * q^89 + 110592 * q^93 + 327204 * q^97

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

−1.73205
1.73205

−27.7128

46.0000

166.277

525.000

1.2

27.7128

46.0000

−166.277

525.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	32.6.a.d	✓	2
3.b	odd	2	1		288.6.a.l		2
4.b	odd	2	1	inner	32.6.a.d	✓	2
5.b	even	2	1		800.6.a.k		2
5.c	odd	4	2		800.6.c.d		4
8.b	even	2	1		64.6.a.h		2
8.d	odd	2	1		64.6.a.h		2
12.b	even	2	1		288.6.a.l		2
16.e	even	4	2		256.6.b.l		4
16.f	odd	4	2		256.6.b.l		4
20.d	odd	2	1		800.6.a.k		2
20.e	even	4	2		800.6.c.d		4
24.f	even	2	1		576.6.a.bp		2
24.h	odd	2	1		576.6.a.bp		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
32.6.a.d	✓	2	1.a	even	1	1	trivial
32.6.a.d	✓	2	4.b	odd	2	1	inner
64.6.a.h		2	8.b	even	2	1
64.6.a.h		2	8.d	odd	2	1
256.6.b.l		4	16.e	even	4	2
256.6.b.l		4	16.f	odd	4	2
288.6.a.l		2	3.b	odd	2	1
288.6.a.l		2	12.b	even	2	1
576.6.a.bp		2	24.f	even	2	1
576.6.a.bp		2	24.h	odd	2	1
800.6.a.k		2	5.b	even	2	1
800.6.a.k		2	20.d	odd	2	1
800.6.c.d		4	5.c	odd	4	2
800.6.c.d		4	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} - 768$ T3^2 - 768 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(32))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 768$ T^2 - 768
$5$	$(T - 46)^{2}$ (T - 46)^2
$7$	$T^{2} - 27648$ T^2 - 27648
$11$	$T^{2} - 6912$ T^2 - 6912
$13$	$(T + 42)^{2}$ (T + 42)^2
$17$	$(T - 962)^{2}$ (T - 962)^2
$19$	$T^{2} - 4320000$ T^2 - 4320000
$23$	$T^{2} - 9980928$ T^2 - 9980928
$29$	$(T + 2554)^{2}$ (T + 2554)^2
$31$	$T^{2} - 3981312$ T^2 - 3981312
$37$	$(T - 11950)^{2}$ (T - 11950)^2
$41$	$(T + 5078)^{2}$ (T + 5078)^2
$43$	$T^{2} - 157600512$ T^2 - 157600512
$47$	$T^{2} - 151400448$ T^2 - 151400448
$53$	$(T + 19714)^{2}$ (T + 19714)^2
$59$	$T^{2} - 79135488$ T^2 - 79135488
$61$	$(T - 29318)^{2}$ (T - 29318)^2
$67$	$T^{2} - 284836608$ T^2 - 284836608
$71$	$T^{2} - 6557248512$ T^2 - 6557248512
$73$	$(T - 37914)^{2}$ (T - 37914)^2
$79$	$T^{2} - 7883993088$ T^2 - 7883993088
$83$	$T^{2} - 1546414848$ T^2 - 1546414848
$89$	$(T - 13930)^{2}$ (T - 13930)^2
$97$	$(T - 163602)^{2}$ (T - 163602)^2
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