LMFDB - Newform orbit 120.6.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,6,Mod(49,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("120.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$120 = 2^{3} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	120.f (of order $2$ , degree $1$ , 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:	$19.2460583776$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.25787221056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 61x^{4} + 852x^{2} + 576$ x^6 + 61x^4 + 852x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}\cdot 5^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$ .

$n$	$31$	$41$	$61$	$97$
$\chi(n)$	$1$	$1$	$1$	$-1$

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}

49.1

	−	4.49053i
		6.33429i
	−	0.843753i
		4.49053i
	−	6.33429i
		0.843753i

−

9.00000i

−51.5439

21.6386i

−

21.3742i

−81.0000

49.2

−

9.00000i

33.8706

44.4723i

85.8595i

−81.0000

49.3

−

9.00000i

42.6733

−

36.1108i

−

168.485i

−81.0000

49.4

9.00000i

−51.5439

−

21.6386i

21.3742i

−81.0000

49.5

9.00000i

33.8706

−

44.4723i

−

85.8595i

−81.0000

49.6

9.00000i

42.6733

36.1108i

168.485i

−81.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	120.6.f.a	✓	6
3.b	odd	2	1		360.6.f.a		6
4.b	odd	2	1		240.6.f.e		6
5.b	even	2	1	inner	120.6.f.a	✓	6
5.c	odd	4	1		600.6.a.r		3
5.c	odd	4	1		600.6.a.s		3
12.b	even	2	1		720.6.f.l		6
15.d	odd	2	1		360.6.f.a		6
20.d	odd	2	1		240.6.f.e		6
60.h	even	2	1		720.6.f.l		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.6.f.a	✓	6	1.a	even	1	1	trivial
120.6.f.a	✓	6	5.b	even	2	1	inner
240.6.f.e		6	4.b	odd	2	1
240.6.f.e		6	20.d	odd	2	1
360.6.f.a		6	3.b	odd	2	1
360.6.f.a		6	15.d	odd	2	1
600.6.a.r		3	5.c	odd	4	1
600.6.a.s		3	5.c	odd	4	1
720.6.f.l		6	12.b	even	2	1
720.6.f.l		6	60.h	even	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000$ T7^6 + 36216*T7^4 + 225603600*T7^2 + 95604640000 acting on $S_{6}^{\mathrm{new}}(120, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$(T^{2} + 81)^{3}$ (T^2 + 81)^3
$5$	$T^{6} + \cdots + 30517578125$ T^6 - 50T^5 - 625T^4 + 283500T^3 - 1953125T^2 - 488281250*T + 30517578125
$7$	$T^{6} + \cdots + 95604640000$ T^6 + 36216T^4 + 225603600T^2 + 95604640000
$11$	$(T^{3} + 332 T^{2} + \cdots + 751600)^{2}$ (T^3 + 332T^2 - 74644T + 751600)^2
$13$	$T^{6} + \cdots + 743071584256$ T^6 + 62184T^4 + 589213584T^2 + 743071584256
$17$	$T^{6} + \cdots + 31\!\cdots\!04$ T^6 + 2125752T^4 + 1451088964368T^2 + 317343789475893504
$19$	$(T^{3} - 144 T^{2} + \cdots + 2584131584)^{2}$ (T^3 - 144T^2 - 5516736T + 2584131584)^2
$23$	$T^{6} + \cdots + 21\!\cdots\!24$ T^6 + 25770672T^4 + 161906214851328T^2 + 219321243807410458624
$29$	$(T^{3} + 946 T^{2} + \cdots + 37303134464)^{2}$ (T^3 + 946T^2 - 25116016T + 37303134464)^2
$31$	$(T^{3} + 5124 T^{2} + \cdots - 71486323200)^{2}$ (T^3 + 5124T^2 - 24591456T - 71486323200)^2
$37$	$T^{6} + \cdots + 54\!\cdots\!36$ T^6 + 125310888T^4 + 1399230201333648T^2 + 543710047431710811136
$41$	$(T^{3} + 662 T^{2} + \cdots + 289547160040)^{2}$ (T^3 + 662T^2 - 204057124T + 289547160040)^2
$43$	$T^{6} + \cdots + 40\!\cdots\!96$ T^6 + 379478448T^4 + 26438712154829568T^2 + 40900336807983666794496
$47$	$T^{6} + \cdots + 29\!\cdots\!00$ T^6 + 1087347984T^4 + 324961555747737600T^2 + 29537257232324005724160000
$53$	$T^{6} + \cdots + 15\!\cdots\!00$ T^6 + 545716248T^4 + 76043629740728976T^2 + 1521825969236124812089600
$59$	$(T^{3} + \cdots - 6990217203600)^{2}$ (T^3 + 5148T^2 - 1547267124T - 6990217203600)^2
$61$	$(T^{3} + \cdots - 30927698302664)^{2}$ (T^3 + 26058T^2 - 1647700020T - 30927698302664)^2
$67$	$T^{6} + \cdots + 34\!\cdots\!56$ T^6 + 593605008T^4 + 18433716615026688T^2 + 34599605672694927917056
$71$	$(T^{3} + \cdots - 37186025753600)^{2}$ (T^3 + 14144T^2 - 3166310416T - 37186025753600)^2
$73$	$T^{6} + \cdots + 20\!\cdots\!00$ T^6 + 10300131936T^4 + 30592485606809913600T^2 + 20843418869900032154920960000
$79$	$(T^{3} + \cdots + 4303737800704)^{2}$ (T^3 - 110100T^2 + 2797561248T + 4303737800704)^2
$83$	$T^{6} + \cdots + 33\!\cdots\!04$ T^6 + 17756024496T^4 + 77417254839780397824T^2 + 3388959104822480512639995904
$89$	$(T^{3} + \cdots + 11306161037592)^{2}$ (T^3 - 175710T^2 + 7437922092T + 11306161037592)^2
$97$	$T^{6} + \cdots + 34\!\cdots\!24$ T^6 + 33401863872T^4 + 270340924137525424128T^2 + 349913889756469630801784602624
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$f(q)$	$=$	$q + 9 \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} - 9 \beta_1 + 8) q^{5} + ( - 3 \beta_{5} + 3 \beta_{4} + \cdots + 38 \beta_1) q^{7} - 81 q^{9} + (9 \beta_{5} + 9 \beta_{4} + \cdots - 102) q^{11} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots - 30 \beta_1) q^{13}+ \cdots + ( - 729 \beta_{5} - 729 \beta_{4} + \cdots + 8262) q^{99}+O(q^{100})$ q + 9b1 q^3 + (-2b5 + b3 - 9b1 + 8) * q^5 + (-3b5 + 3b4 + 2b3 + 2b2 + 38b1) q^7 - 81 * q^9 + (9b5 + 9b4 + 4b3 - 4b2 - 102) * q^11 + (-2b5 + 2b4 - 3b3 - 3b2 - 30b1) q^13 + (-9b5 - 18b3 + 72b1 + 81) q^15 + (16b5 - 16b4 - 23b3 - 23b2 - 398b1) q^17 + (64b5 + 64b4 + 20b3 - 20b2 + 104) * q^19 + (-18b5 - 18b4 - 27b3 + 27b2 - 342) * q^21 + (-80b5 + 80b4 + 52b3 + 52b2 - 868b1) q^23 + (-20b5 + 100b4 + 85b3 + 75b2 + 860b1 + 655) q^25 - 729b1 q^27 + (-6b5 - 6b4 - 115b3 + 115b2 - 396) * q^29 + (66b5 + 66b4 - 96b3 + 96b2 - 1728) * q^31 + (-36b5 + 36b4 + 81b3 + 81b2 - 918b1) q^33 + (-85b5 + 175b4 + 100b2 + 1180b1 - 3150) * q^35 + (-74b5 + 74b4 + 197b3 + 197b2 - 2422b1) q^37 + (27b5 + 27b4 - 18b3 + 18b2 + 270) * q^39 + (388b5 + 388b4 + 102b3 - 102b2 + 106) * q^41 + (-180b5 + 180b4 - 60b3 - 60b2 - 8316b1) q^43 + (162b5 - 81b3 + 729b1 - 648) q^45 + (-286b5 + 286b4 - 316b3 - 316b2 + 8456b1) q^47 + (24b5 + 24b4 - 318b3 + 318b2 + 4539) * q^49 + (207b5 + 207b4 + 144b3 - 144b2 + 3582) * q^51 + (-142b5 + 142b4 - 25b3 - 25b2 - 12230b1) q^53 + (337b5 - 25b4 - 166b3 - 550b2 + 5954b1 - 7628) q^55 + (-180b5 + 180b4 + 576b3 + 576b2 + 936b1) q^57 + (643b5 + 643b4 + 920b3 - 920b2 - 674) * q^59 + (-904b5 - 904b4 - 920b3 + 920b2 - 9902) * q^61 + (243b5 - 243b4 - 162b3 - 162b2 - 3078b1) q^63 + (-88b5 - 100b4 + 89b3 + 175b2 + 5804b1 - 2338) q^65 + (-286b5 + 286b4 + 112b3 + 112b2 + 10480b1) q^67 + (-468b5 - 468b4 - 720b3 + 720b2 + 7812) * q^69 + (-366b5 - 366b4 + 1120b3 - 1120b2 - 4212) * q^71 + (980b5 - 980b4 + 1038b3 + 1038b2 - 12540b1) q^73 + (-765b5 - 675b4 - 180b3 + 900b2 + 5895b1 - 7740) q^75 + (376b5 - 376b4 - 578b3 - 578b2 + 14964b1) q^77 + (-470b5 - 470b4 + 536b3 - 536b2 + 36744) * q^79 + 6561 * q^81 + (260b5 - 260b4 - 2476b3 - 2476b2 - 4364b1) q^83 + (402b5 - 1550b4 + 329b3 - 225b2 + 7684b1 + 15782) q^85 + (1035b5 - 1035b4 - 54b3 - 54b2 - 3564b1) q^87 + (-1296b5 - 1296b4 + 120b3 - 120b2 + 57786) * q^89 + (-38b5 - 38b4 - 280b3 + 280b2 - 3060) * q^91 + (864b5 - 864b4 + 594b3 + 594b2 - 15552b1) q^93 + (552b5 - 600b4 - 596b3 - 3700b2 + 25384b1 - 57168) q^95 + (2828b5 - 2828b4 + 204b3 + 204b2 - 3960b1) q^97 + (-729b5 - 729b4 - 324b3 + 324b2 + 8262) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51}+ \cdots + 53784 q^{99}+O(q^{100})$ 6 * q + 50 * q^5 - 486 * q^9 - 664 * q^11 + 540 * q^15 + 288 * q^19 - 1872 * q^21 + 3750 * q^25 - 1892 * q^29 - 10248 * q^31 - 18880 * q^35 + 1584 * q^39 - 1324 * q^41 - 4050 * q^45 + 28410 * q^49 + 20088 * q^51 - 47160 * q^55 - 10296 * q^59 - 52116 * q^61 - 13480 * q^65 + 51624 * q^69 - 28288 * q^71 - 41400 * q^75 + 220200 * q^79 + 39366 * q^81 + 95880 * q^85 + 351420 * q^89 - 17088 * q^91 - 349120 * q^95 + 53784 * q^99

$\beta_{1}$	$=$	$( \nu^{5} + 85\nu^{3} + 1596\nu ) / 1296$ (v^5 + 85v^3 + 1596v) / 1296
$\beta_{2}$	$=$	$( -7\nu^{5} - 28\nu^{4} - 379\nu^{3} - 1516\nu^{2} - 2748\nu - 11856 ) / 432$ (-7v^5 - 28v^4 - 379v^3 - 1516v^2 - 2748*v - 11856) / 432
$\beta_{3}$	$=$	$( -7\nu^{5} + 28\nu^{4} - 379\nu^{3} + 1516\nu^{2} - 2748\nu + 11856 ) / 432$ (-7v^5 + 28v^4 - 379v^3 + 1516v^2 - 2748*v + 11856) / 432
$\beta_{4}$	$=$	$( 65\nu^{5} - 108\nu^{4} + 3581\nu^{3} - 3996\nu^{2} + 40884\nu - 9072 ) / 1296$ (65v^5 - 108v^4 + 3581v^3 - 3996v^2 + 40884*v - 9072) / 1296
$\beta_{5}$	$=$	$( -65\nu^{5} - 108\nu^{4} - 3581\nu^{3} - 3996\nu^{2} - 40884\nu - 9072 ) / 1296$ (-65v^5 - 108v^4 - 3581v^3 - 3996v^2 - 40884*v - 9072) / 1296

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	120.6.f.a

Level	$120$
Weight	$6$
Character orbit	120.f
Analytic conductor	$19.246$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

$\nu$	$=$	$( -\beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4\beta_1 ) / 20$ (-b5 + b4 + 3b3 + 3b2 - 4*b1) / 20
$\nu^{2}$	$=$	$( 7\beta_{5} + 7\beta_{4} + 9\beta_{3} - 9\beta_{2} - 396 ) / 20$ (7b5 + 7b4 + 9b3 - 9b2 - 396) / 20
$\nu^{3}$	$=$	$( 39\beta_{5} - 39\beta_{4} - 97\beta_{3} - 97\beta_{2} + 996\beta_1 ) / 20$ (39b5 - 39b4 - 97b3 - 97b2 + 996*b1) / 20
$\nu^{4}$	$=$	$( -379\beta_{5} - 379\beta_{4} - 333\beta_{3} + 333\beta_{2} + 12972 ) / 20$ (-379b5 - 379b4 - 333b3 + 333b2 + 12972) / 20
$\nu^{5}$	$=$	$( -1719\beta_{5} + 1719\beta_{4} + 3457\beta_{3} + 3457\beta_{2} - 52356\beta_1 ) / 20$ (-1719b5 + 1719b4 + 3457b3 + 3457b2 - 52356*b1) / 20

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.6
Significant digits:
Format: