LMFDB - Newform orbit 720.6.f.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, 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("720.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, 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:	$115.476350265$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 9) q^{5}+ \cdots + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots + 7 \beta_1) q^{7}+O(q^{10}) $ q + (-b5 - b4 - b2 - 3b1 - 9) q^5 + (-2b4 + 2b3 - 3b2 + 7b1) * q^7	$ q + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 9) q^{5}+ \cdots + (204 \beta_{4} - 204 \beta_{3} + \cdots - 322 \beta_1) q^{97}+O(q^{100}) $ q + (-b5 - b4 - b2 - 3b1 - 9) q^5 + (-2b4 + 2b3 - 3b2 + 7b1) * q^7 + (-9b5 + 4b4 + 4b3 - 111) q^11 + (-3b4 + 3b3 + 2b2 + 7b1) * q^13 + (23b4 - 23b3 + 16b2 - 80b1) * q^17 + (64b5 - 20b4 - 20b3 - 40) q^19 + (52b4 - 52b3 + 80b2 + 283b1) * q^23 + (-40b5 + 85b4 - 75b3 + 60b2 - 145b1 + 615) q^25 + (-6b5 + 115b4 + 115b3 + 390) q^29 + (66b5 + 96b4 + 96b3 + 1794) q^31 + (-45b5 - 100b3 + 130b2 - 205b1 - 3195) * q^35 + (197b4 - 197b3 + 74b2 + 741b1) * q^37 + (388b5 - 102b4 - 102b3 + 282) q^41 + (60b4 - 60b3 - 180b2 - 2139b1) * q^43 + (-316b4 + 316b3 + 286b2 - 2129b1) * q^47 + (-24b5 - 318b4 - 318b3 + 4515) q^49 + (25b4 - 25b3 - 142b2 - 3116b1) * q^53 + (156b5 + 166b4 - 550b3 + 181b2 + 1758b1 + 7784) q^55 + (-643b5 + 920b4 + 920b3 - 1317) q^59 + (904b5 - 920b4 - 920b3 - 8998) q^61 + (-94b5 - 89b4 + 175b3 + 6b2 + 1388b1 + 2244) q^65 + (-112b4 + 112b3 - 286b2 + 2421b1) * q^67 + (366b5 + 1120b4 + 1120b3 - 3846) q^71 + (1038b4 - 1038b3 - 980b2 + 3164b1) * q^73 + (578b4 - 578b3 + 376b2 + 4218b1) * q^77 + (-470b5 - 536b4 - 536b3 - 37214) q^79 + (-2476b4 + 2476b3 - 260b2 - 277b1) * q^83 + (574b5 + 329b4 + 225b3 - 976b2 - 2383b1 + 16356) q^85 + (-1296b5 - 120b4 - 120b3 - 59082) q^89 + (-38b5 + 280b4 + 280b3 + 3022) q^91 + (24b5 - 596b4 + 3700b3 - 576b2 - 7708b1 - 57144) q^95 + (204b4 - 204b3 - 2828b2 - 322b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 50 q^{5}+O(q^{10}) $ 6 * q - 50 * q^5	$ 6 q - 50 q^{5} - 664 q^{11} - 288 q^{19} + 3750 q^{25} + 1892 q^{29} + 10248 q^{31} - 18880 q^{35} + 1324 q^{41} + 28410 q^{49} + 47160 q^{55} - 10296 q^{59} - 52116 q^{61} + 13480 q^{65} - 28288 q^{71} - 220200 q^{79} + 95880 q^{85} - 351420 q^{89} + 17088 q^{91} - 349120 q^{95}+O(q^{100}) $ 6 * q - 50 * q^5 - 664 * q^11 - 288 * q^19 + 3750 * q^25 + 1892 * q^29 + 10248 * q^31 - 18880 * q^35 + 1324 * q^41 + 28410 * q^49 + 47160 * q^55 - 10296 * q^59 - 52116 * q^61 + 13480 * q^65 - 28288 * q^71 - 220200 * q^79 + 95880 * q^85 - 351420 * q^89 + 17088 * q^91 - 349120 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 61x^{4} + 852x^{2} + 576 $ x^6 + 61*x^4 + 852*x^2 + 576 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 85\nu^{3} + 1596\nu ) / 324 $ (v^5 + 85v^3 + 1596v) / 324
$\beta_{2}$	$=$	$ ( -11\nu^{5} - 611\nu^{3} - 7080\nu ) / 108 $ (-11v^5 - 611v^3 - 7080*v) / 108
$\beta_{3}$	$=$	$ ( -5\nu^{5} + 21\nu^{4} - 263\nu^{3} + 1137\nu^{2} - 1662\nu + 8892 ) / 324 $ (-5v^5 + 21v^4 - 263v^3 + 1137v^2 - 1662*v + 8892) / 324
$\beta_{4}$	$=$	$ ( 5\nu^{5} + 21\nu^{4} + 263\nu^{3} + 1137\nu^{2} + 1662\nu + 8892 ) / 324 $ (5v^5 + 21v^4 + 263v^3 + 1137v^2 + 1662*v + 8892) / 324
$\beta_{5}$	$=$	$ ( \nu^{4} + 37\nu^{2} + 78 ) / 6 $ (v^4 + 37*v^2 + 78) / 6

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -3\beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta_1 ) / 20 $ (-3b4 + 3b3 - b2 - 3*b1) / 20
$\nu^{2}$	$=$	$ ( -7\beta_{5} + 9\beta_{4} + 9\beta_{3} - 403 ) / 20 $ (-7b5 + 9b4 + 9*b3 - 403) / 20
$\nu^{3}$	$=$	$ ( 97\beta_{4} - 97\beta_{3} + 39\beta_{2} + 317\beta_1 ) / 20 $ (97b4 - 97b3 + 39b2 + 317b1) / 20
$\nu^{4}$	$=$	$ ( 379\beta_{5} - 333\beta_{4} - 333\beta_{3} + 13351 ) / 20 $ (379b5 - 333b4 - 333*b3 + 13351) / 20
$\nu^{5}$	$=$	$ ( -3457\beta_{4} + 3457\beta_{3} - 1719\beta_{2} - 15677\beta_1 ) / 20 $ (-3457b4 + 3457b3 - 1719b2 - 15677b1) / 20

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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} $

289.1

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

−42.6733

−

36.1108i

−

168.485i

289.2

−42.6733

36.1108i

168.485i

289.3

−33.8706

−

44.4723i

−

85.8595i

289.4

−33.8706

44.4723i

85.8595i

289.5

51.5439

−

21.6386i

21.3742i

289.6

51.5439

21.6386i

−

21.3742i

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	720.6.f.l		6
3.b	odd	2	1		240.6.f.e		6
4.b	odd	2	1		360.6.f.a		6
5.b	even	2	1	inner	720.6.f.l		6
12.b	even	2	1		120.6.f.a	✓	6
15.d	odd	2	1		240.6.f.e		6
20.d	odd	2	1		360.6.f.a		6
60.h	even	2	1		120.6.f.a	✓	6
60.l	odd	4	1		600.6.a.r		3
60.l	odd	4	1		600.6.a.s		3

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(720, [\chi])$:

$ T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000 $

$ T_{11}^{3} + 332T_{11}^{2} - 74644T_{11} + 751600 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ 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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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 9) q^{5}+ \cdots + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots + 7 \beta_1) q^{7}+O(q^{10}) \) q + (-b5 - b4 - b2 - 3b1 - 9) q^5 + (-2b4 + 2b3 - 3b2 + 7b1) * q^7	\( q + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 9) q^{5}+ \cdots + (204 \beta_{4} - 204 \beta_{3} + \cdots - 322 \beta_1) q^{97}+O(q^{100}) \) q + (-b5 - b4 - b2 - 3b1 - 9) q^5 + (-2b4 + 2b3 - 3b2 + 7b1) * q^7 + (-9b5 + 4b4 + 4b3 - 111) q^11 + (-3b4 + 3b3 + 2b2 + 7b1) * q^13 + (23b4 - 23b3 + 16b2 - 80b1) * q^17 + (64b5 - 20b4 - 20b3 - 40) q^19 + (52b4 - 52b3 + 80b2 + 283b1) * q^23 + (-40b5 + 85b4 - 75b3 + 60b2 - 145b1 + 615) q^25 + (-6b5 + 115b4 + 115b3 + 390) q^29 + (66b5 + 96b4 + 96b3 + 1794) q^31 + (-45b5 - 100b3 + 130b2 - 205b1 - 3195) * q^35 + (197b4 - 197b3 + 74b2 + 741b1) * q^37 + (388b5 - 102b4 - 102b3 + 282) q^41 + (60b4 - 60b3 - 180b2 - 2139b1) * q^43 + (-316b4 + 316b3 + 286b2 - 2129b1) * q^47 + (-24b5 - 318b4 - 318b3 + 4515) q^49 + (25b4 - 25b3 - 142b2 - 3116b1) * q^53 + (156b5 + 166b4 - 550b3 + 181b2 + 1758b1 + 7784) q^55 + (-643b5 + 920b4 + 920b3 - 1317) q^59 + (904b5 - 920b4 - 920b3 - 8998) q^61 + (-94b5 - 89b4 + 175b3 + 6b2 + 1388b1 + 2244) q^65 + (-112b4 + 112b3 - 286b2 + 2421b1) * q^67 + (366b5 + 1120b4 + 1120b3 - 3846) q^71 + (1038b4 - 1038b3 - 980b2 + 3164b1) * q^73 + (578b4 - 578b3 + 376b2 + 4218b1) * q^77 + (-470b5 - 536b4 - 536b3 - 37214) q^79 + (-2476b4 + 2476b3 - 260b2 - 277b1) * q^83 + (574b5 + 329b4 + 225b3 - 976b2 - 2383b1 + 16356) q^85 + (-1296b5 - 120b4 - 120b3 - 59082) q^89 + (-38b5 + 280b4 + 280b3 + 3022) q^91 + (24b5 - 596b4 + 3700b3 - 576b2 - 7708b1 - 57144) q^95 + (204b4 - 204b3 - 2828b2 - 322b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 50 q^{5}+O(q^{10}) \) 6 * q - 50 * q^5	\( 6 q - 50 q^{5} - 664 q^{11} - 288 q^{19} + 3750 q^{25} + 1892 q^{29} + 10248 q^{31} - 18880 q^{35} + 1324 q^{41} + 28410 q^{49} + 47160 q^{55} - 10296 q^{59} - 52116 q^{61} + 13480 q^{65} - 28288 q^{71} - 220200 q^{79} + 95880 q^{85} - 351420 q^{89} + 17088 q^{91} - 349120 q^{95}+O(q^{100}) \) 6 * q - 50 * q^5 - 664 * q^11 - 288 * q^19 + 3750 * q^25 + 1892 * q^29 + 10248 * q^31 - 18880 * q^35 + 1324 * q^41 + 28410 * q^49 + 47160 * q^55 - 10296 * q^59 - 52116 * q^61 + 13480 * q^65 - 28288 * q^71 - 220200 * q^79 + 95880 * q^85 - 351420 * q^89 + 17088 * q^91 - 349120 * q^95

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