LMFDB - Newform orbit 400.5.p.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,5,Mod(193,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.193");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	400.p (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$41.3479852335$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{61})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 31x^{2} + 225$ x^4 + 31*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$\beta_{1}$	$1$

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}

193.1

		4.40512i
	−	3.40512i
	−	4.40512i
		3.40512i

−11.8102

−

11.8102i

19.8102

−

19.8102i

197.964i

193.2

3.81025

3.81025i

4.18975

−

4.18975i

−

51.9640i

257.1

−11.8102

11.8102i

19.8102

19.8102i

−

197.964i

257.2

3.81025

−

3.81025i

4.18975

4.18975i

51.9640i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.5.p.f		4
4.b	odd	2	1		200.5.l.d	yes	4
5.b	even	2	1		400.5.p.m		4
5.c	odd	4	1	inner	400.5.p.f		4
5.c	odd	4	1		400.5.p.m		4
20.d	odd	2	1		200.5.l.b	✓	4
20.e	even	4	1		200.5.l.b	✓	4
20.e	even	4	1		200.5.l.d	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.5.l.b	✓	4	20.d	odd	2	1
200.5.l.b	✓	4	20.e	even	4	1
200.5.l.d	yes	4	4.b	odd	2	1
200.5.l.d	yes	4	20.e	even	4	1
400.5.p.f		4	1.a	even	1	1	trivial
400.5.p.f		4	5.c	odd	4	1	inner
400.5.p.m		4	5.b	even	2	1
400.5.p.m		4	5.c	odd	4	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} + 16T_{3}^{3} + 128T_{3}^{2} - 1440T_{3} + 8100$ T3^4 + 16*T3^3 + 128*T3^2 - 1440*T3 + 8100 acting on $S_{5}^{\mathrm{new}}(400, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 16 T^{3} + \cdots + 8100$ T^4 + 16T^3 + 128T^2 - 1440*T + 8100
$5$	$T^{4}$ T^4
$7$	$T^{4} - 48 T^{3} + \cdots + 27556$ T^4 - 48T^3 + 1152T^2 - 7968*T + 27556
$11$	$(T^{2} - 120 T - 12016)^{2}$ (T^2 - 120*T - 12016)^2
$13$	$T^{4} + 384 T^{3} + \cdots + 746496$ T^4 + 384T^3 + 73728T^2 + 331776*T + 746496
$17$	$T^{4} + 512 T^{3} + \cdots + 231040000$ T^4 + 512T^3 + 131072T^2 + 7782400*T + 231040000
$19$	$T^{4} + \cdots + 17041735936$ T^4 + 301088*T^2 + 17041735936
$23$	$T^{4} + \cdots + 155067413796$ T^4 - 144T^3 + 10368T^2 + 56705184*T + 155067413796
$29$	$T^{4} + \cdots + 51690750736$ T^4 + 544712*T^2 + 51690750736
$31$	$(T^{2} + 1552 T + 40000)^{2}$ (T^2 + 1552*T + 40000)^2
$37$	$T^{4} + \cdots + 97863860224$ T^4 + 3840T^3 + 7372800T^2 + 1201274880*T + 97863860224
$41$	$(T^{2} + 144 T - 1884352)^{2}$ (T^2 + 144*T - 1884352)^2
$43$	$T^{4} + \cdots + 1398083948836$ T^4 - 3792T^3 + 7189632T^2 - 4483683552*T + 1398083948836
$47$	$T^{4} + \cdots + 82145183306404$ T^4 - 8528T^3 + 36363392T^2 - 77292658144*T + 82145183306404
$53$	$T^{4} + \cdots + 13008198329344$ T^4 + 1664T^3 + 1384448T^2 - 6001528832*T + 13008198329344
$59$	$T^{4} + \cdots + 8392655352064$ T^4 + 27249440*T^2 + 8392655352064
$61$	$(T^{2} + 5472 T + 6923520)^{2}$ (T^2 + 5472*T + 6923520)^2
$67$	$T^{4} + \cdots + 81224687600676$ T^4 + 1968T^3 + 1936512T^2 - 17736548832*T + 81224687600676
$71$	$(T^{2} + 8928 T + 19677440)^{2}$ (T^2 + 8928*T + 19677440)^2
$73$	$T^{4} + \cdots + 111331664896$ T^4 + 2304T^3 + 2654208T^2 + 768761856*T + 111331664896
$79$	$T^{4} + \cdots + 17\!\cdots\!00$ T^4 + 98630784*T^2 + 1743657070694400
$83$	$T^{4} + \cdots + 560151218940196$ T^4 + 688T^3 + 236672T^2 - 16283249632*T + 560151218940196
$89$	$T^{4} + \cdots + 37\!\cdots\!76$ T^4 + 126503048*T^2 + 3751890317160976
$97$	$T^{4} + \cdots + 38\!\cdots\!44$ T^4 - 8448T^3 + 35684352T^2 + 1666443288576*T + 38911078363423744
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Abelian varieties over $\F_{q}$

$f(q)$	$=$	$q + ( - \beta_{3} + 4 \beta_1 - 4) q^{3} + ( - \beta_{2} + 12 \beta_1 + 12) q^{7} + (8 \beta_{3} + 8 \beta_{2} - 73 \beta_1) q^{9} + (8 \beta_{3} - 8 \beta_{2} + 60) q^{11} + ( - 12 \beta_{3} + 96 \beta_1 - 96) q^{13}+ \cdots + (1064 \beta_{3} + 1064 \beta_{2} - 19996 \beta_1) q^{99}+O(q^{100})$ q + (-b3 + 4b1 - 4) q^3 + (-b2 + 12b1 + 12) q^7 + (8b3 + 8b2 - 73b1) q^9 + (8b3 - 8b2 + 60) * q^11 + (-12b3 + 96b1 - 96) * q^13 + (-12b2 - 128b1 - 128) * q^17 + (-24b3 - 24b2 + 100b1) q^19 + (-16b3 + 16b2 - 218) * q^21 + (-57b3 - 36b1 + 36) * q^23 + (-56b2 + 944b1 + 944) * q^27 + (-32b3 - 32b2 - 150b1) q^29 + (-48b3 + 48b2 - 776) * q^31 + (-124b3 + 1216b1 - 1216) * q^33 + (-112b2 - 960b1 - 960) * q^37 + (144b3 + 144b2 - 2232b1) q^39 + (88b3 - 88b2 - 72) * q^41 + (-71b3 - 948b1 + 948) * q^43 + (-15b2 + 2132b1 + 2132) * q^47 + (-24b3 - 24b2 - 1991b1) q^49 + (80b3 - 80b2 - 440) * q^51 + (180b3 + 416b1 - 416) * q^53 + (292b2 - 3328b1 - 3328) * q^57 + (-184b3 - 184b2 - 2316b1) q^59 + (-48b3 + 48b2 - 2736) * q^61 + (265b3 - 1852b1 + 1852) * q^63 + (279b2 - 492b1 - 492) * q^67 + (192b3 + 192b2 - 6666b1) q^69 + (32b3 - 32b2 - 4464) * q^71 + (52b3 + 576b1 - 576) * q^73 + (-252b2 + 1696b1 + 1696) * q^77 + (-432b3 - 432b2 - 1944b1) q^79 + (-520b3 + 520b2 - 8471) * q^81 + (441b3 + 172b1 - 172) * q^83 + (106b2 - 3304b1 - 3304) * q^87 + (64b3 + 64b2 - 7890b1) q^89 + (-240b3 + 240b2 - 3768) * q^91 + (1160b3 - 8960b1 + 8960) * q^93 + (1300b2 + 2112b1 + 2112) * q^97 + (1064b3 + 1064b2 - 19996b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 16 q^{3} + 48 q^{7} + 240 q^{11} - 384 q^{13} - 512 q^{17} - 872 q^{21} + 144 q^{23} + 3776 q^{27} - 3104 q^{31} - 4864 q^{33} - 3840 q^{37} - 288 q^{41} + 3792 q^{43} + 8528 q^{47} - 1760 q^{51}+ \cdots + 8448 q^{97}+O(q^{100})$ 4 * q - 16 * q^3 + 48 * q^7 + 240 * q^11 - 384 * q^13 - 512 * q^17 - 872 * q^21 + 144 * q^23 + 3776 * q^27 - 3104 * q^31 - 4864 * q^33 - 3840 * q^37 - 288 * q^41 + 3792 * q^43 + 8528 * q^47 - 1760 * q^51 - 1664 * q^53 - 13312 * q^57 - 10944 * q^61 + 7408 * q^63 - 1968 * q^67 - 17856 * q^71 - 2304 * q^73 + 6784 * q^77 - 33884 * q^81 - 688 * q^83 - 13216 * q^87 - 15072 * q^91 + 35840 * q^93 + 8448 * q^97

$\beta_{1}$	$=$	$( \nu^{3} + 16\nu ) / 15$ (v^3 + 16*v) / 15
$\beta_{2}$	$=$	$( \nu^{3} + 30\nu^{2} + 46\nu + 465 ) / 15$ (v^3 + 30v^2 + 46v + 465) / 15
$\beta_{3}$	$=$	$( \nu^{3} - 30\nu^{2} + 46\nu - 465 ) / 15$ (v^3 - 30v^2 + 46v - 465) / 15

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