LMFDB - Newform orbit 252.9.d.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(181,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.181");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.d (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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 2160x^{4} + 976392x^{2} + 85162752 $ x^6 + 2160x^4 + 976392x^2 + 85162752
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 28)
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_1 q^{5} + ( - \beta_{3} + 361) q^{7}+O(q^{10}) $ q - b1 * q^5 + (-b3 + 361) * q^7	$ q - \beta_1 q^{5} + ( - \beta_{3} + 361) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_1 - 4082) q^{11} + ( - \beta_{4} - 6 \beta_{3} + \cdots - 29 \beta_1) q^{13}+ \cdots + ( - 637 \beta_{4} - 3822 \beta_{3} + \cdots - 28328 \beta_1) q^{97}+O(q^{100}) $ q - b1 * q^5 + (-b3 + 361) * q^7 + (b5 - 2b3 - b1 - 4082) q^11 + (-b4 - 6b3 - 3b2 - 29b1) q^13 + (3b4 + 18b3 + 45b2 - 100b1) * q^17 + (-5b4 - 30b3 + 50b2 - 64b1) * q^19 + (5b5 + 6b4 - 58b3 + 6b2 - 29b1 + 1934) q^23 + (15b5 - 11b4 + 58b3 - 11b2 + 29b1 - 113119) q^25 + (22b5 - 30b4 + 196b3 - 30b2 + 98b1 - 210722) q^29 + (15b4 + 90b3 - 1515b2 + 996b1) * q^31 + (-35b5 - 21b4 - 2b3 - 1596b2 - 189b1 + 219136) q^35 + (30b5 + 78b4 - 684b3 + 78b2 - 342b1 + 530722) q^37 + (-90b4 - 540b3 + 1290b2 - 2078b1) * q^41 + (105b5 - 70b4 + 350b3 - 70b2 + 175b1 - 1297230) q^43 + (81b4 + 486b3 - 4809b2 - 364b1) * q^47 + (-210b5 + 119b4 - 542b3 + 4389b2 - 4172b1 + 453185) q^49 + (-300b5 + 180b4 - 840b3 + 180b2 - 420b1 + 1390110) q^53 + (-375b4 - 2250b3 - 1655b2 + 12948b1) * q^55 + (435b4 + 2610b3 + 5130b2 - 1784b1) * q^59 + (-20b4 - 120b3 + 3300b2 + 10019b1) * q^61 + (35b5 - 483b4 + 3794b3 - 483b2 + 1897b1 - 14015872) q^65 + (-435b5 - 490b4 + 4790b3 - 490b2 + 2395b1 + 2763250) q^67 + (560b5 - 219b4 + 632b3 - 219b2 + 316b1 + 10348142) q^71 + (853b4 + 5118b3 + 33803b2 - 3346b1) * q^73 + (245b5 - 2205b4 + 4418b3 + 28371b2 - 43463b1 + 10231742) q^77 + (834b5 - 471b4 + 2100b3 - 471b2 + 1050b1 + 31019090) q^79 + (-738b4 - 4428b3 - 29163b2 + 43240b1) * q^83 + (2160b5 + 580b4 - 8960b3 + 580b2 - 4480b1 - 43868480) q^85 + (-2445b4 - 14670b3 + 21285b2 - 25554b1) * q^89 + (-2583b5 + 224b4 - 4076b3 - 23401b2 - 39977b1 - 29850176) q^91 + (-2015b5 + 621b4 - 938b3 + 621b2 - 469b1 - 14318816) q^95 + (-637b4 - 3822b3 - 22843b2 - 28328b1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2166 q^{7}+O(q^{10}) $ 6 * q + 2166 * q^7	$ 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95}+O(q^{100}) $ 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 2160x^{4} + 976392x^{2} + 85162752 $ x^6 + 2160*x^4 + 976392*x^2 + 85162752 :

$\beta_{1}$	$=$	$ ( -\nu^{5} - 828\nu^{3} + 5640984\nu ) / 183816 $ (-v^5 - 828v^3 + 5640984v) / 183816
$\beta_{2}$	$=$	$ ( \nu^{5} + 828\nu^{3} - 494136\nu ) / 20424 $ (v^5 + 828v^3 - 494136v) / 20424
$\beta_{3}$	$=$	$ ( 109\nu^{5} - 1332\nu^{4} + 212796\nu^{3} - 2021976\nu^{2} + 63046152\nu - 251332416 ) / 183816 $ (109v^5 - 1332v^4 + 212796v^3 - 2021976v^2 + 63046152*v - 251332416) / 183816
$\beta_{4}$	$=$	$ ( 859\nu^{5} + 7992\nu^{4} + 1691604\nu^{3} + 12131856\nu^{2} + 531380376\nu + 1507994496 ) / 183816 $ (859v^5 + 7992v^4 + 1691604v^3 + 12131856v^2 + 531380376*v + 1507994496) / 183816
$\beta_{5}$	$=$	$ ( 217\nu^{5} + 15984\nu^{4} + 424764\nu^{3} + 31983984\nu^{2} + 131733288\nu + 8574584832 ) / 183816 $ (217v^5 + 15984v^4 + 424764v^3 + 31983984v^2 + 131733288*v + 8574584832) / 183816

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

$\nu$	$=$	$ ( \beta_{2} + 9\beta_1 ) / 252 $ (b2 + 9*b1) / 252
$\nu^{2}$	$=$	$ ( \beta_{5} - \beta_{4} + 6\beta_{3} - \beta_{2} + 3\beta _1 - 30240 ) / 42 $ (b5 - b4 + 6b3 - b2 + 3b1 - 30240) / 42
$\nu^{3}$	$=$	$ ( 9\beta_{4} + 54\beta_{3} - 1835\beta_{2} - 2898\beta_1 ) / 84 $ (9b4 + 54b3 - 1835b2 - 2898b1) / 84
$\nu^{4}$	$=$	$ ( -253\beta_{5} + 322\beta_{4} - 2070\beta_{3} + 322\beta_{2} - 1035\beta _1 + 6329904 ) / 7 $ (-253b5 + 322b4 - 2070b3 + 322b2 - 1035*b1 + 6329904) / 7
$\nu^{5}$	$=$	$ ( -621\beta_{4} - 3726\beta_{3} + 283309\beta_{2} + 323496\beta_1 ) / 7 $ (-621b4 - 3726b3 + 283309b2 + 323496b1) / 7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$1$

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} $

181.1

		39.6746i
		21.7042i
		10.7169i
	−	10.7169i
	−	21.7042i
	−	39.6746i

−

964.066i

2367.93

397.129i

181.2

−

685.916i

−1845.45

1535.94i

181.3

−

333.657i

560.525

−

2334.65i

181.4

333.657i

560.525

2334.65i

181.5

685.916i

−1845.45

−

1535.94i

181.6

964.066i

2367.93

−

397.129i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.9.d.b		6
3.b	odd	2	1		28.9.b.a	✓	6
7.b	odd	2	1	inner	252.9.d.b		6
12.b	even	2	1		112.9.c.d		6
21.c	even	2	1		28.9.b.a	✓	6
21.g	even	6	2		196.9.h.b		12
21.h	odd	6	2		196.9.h.b		12
84.h	odd	2	1		112.9.c.d		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.9.b.a	✓	6	3.b	odd	2	1
28.9.b.a	✓	6	21.c	even	2	1
112.9.c.d		6	12.b	even	2	1
112.9.c.d		6	84.h	odd	2	1
196.9.h.b		12	21.g	even	6	2
196.9.h.b		12	21.h	odd	6	2
252.9.d.b		6	1.a	even	1	1	trivial
252.9.d.b		6	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 1511232T_{5}^{4} + 593123565600T_{5}^{2} + 48680661813120000 $ T5^6 + 1511232*T5^4 + 593123565600*T5^2 + 48680661813120000 acting on $S_{9}^{\mathrm{new}}(252, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + \cdots + 48\!\cdots\!00 $ T^6 + 1511232T^4 + 593123565600T^2 + 48680661813120000
$7$	$ T^{6} + \cdots + 19\!\cdots\!01 $ T^6 - 2166T^5 + 986223T^4 - 5377596532T^3 + 5685379336623T^2 - 71982527613755766*T + 191581231380566414401
$11$	$ (T^{3} + \cdots - 7013477844792)^{2} $ (T^3 + 12246T^2 - 586924020T - 7013477844792)^2
$13$	$ T^{6} + \cdots + 11\!\cdots\!68 $ T^6 + 2691986880T^4 + 1550774551274383392T^2 + 117641902891055469677568
$17$	$ T^{6} + \cdots + 15\!\cdots\!00 $ T^6 + 29101834752T^4 + 131928294002223292416T^2 + 158369212003410132239646720000
$19$	$ T^{6} + \cdots + 20\!\cdots\!00 $ T^6 + 43801462272T^4 + 21924088485169773600T^2 + 2060516417807823110709120000
$23$	$ (T^{3} + \cdots + 60\!\cdots\!00)^{2} $ (T^3 - 5802T^2 - 62186581620T + 6005643700768200)^2
$29$	$ (T^{3} + \cdots + 10\!\cdots\!44)^{2} $ (T^3 + 632166T^2 - 638738692596T + 106390583279134344)^2
$31$	$ T^{6} + \cdots + 44\!\cdots\!00 $ T^6 + 5071197100032T^4 + 8320018694388229656576000T^2 + 4400458684536939958007566958592000000
$37$	$ (T^{3} + \cdots + 98\!\cdots\!00)^{2} $ (T^3 - 1592166T^2 - 6206563348980T + 9891859374136705400)^2
$41$	$ T^{6} + \cdots + 61\!\cdots\!00 $ T^6 + 17909852667648T^4 + 8743524484037918150361600T^2 + 6115998445514110962869305466880000
$43$	$ (T^{3} + \cdots - 76\!\cdots\!00)^{2} $ (T^3 + 3891690T^2 - 2651175088500T - 7677283367599693000)^2
$47$	$ T^{6} + \cdots + 22\!\cdots\!12 $ T^6 + 68017743250944T^4 + 999350529508479666230599680T^2 + 2284747171397913605028538611625950707712
$53$	$ (T^{3} + \cdots + 11\!\cdots\!00)^{2} $ (T^3 - 4170330T^2 - 53350797654900T + 113085812349078885000)^2
$59$	$ T^{6} + \cdots + 36\!\cdots\!00 $ T^6 + 338441040125952T^4 + 30857760741519814342256877600T^2 + 368652700774423333627770053019134977920000
$61$	$ T^{6} + \cdots + 13\!\cdots\!00 $ T^6 + 225118443807552T^4 + 5393030932438337494106157600T^2 + 13158816814061453716309691279689345920000
$67$	$ (T^{3} + \cdots - 22\!\cdots\!00)^{2} $ (T^3 - 8289750T^2 - 412665478211700T - 2221002499578768941000)^2
$71$	$ (T^{3} + \cdots + 22\!\cdots\!40)^{2} $ (T^3 - 31044426T^2 + 139808421637164T + 22965958767033039240)^2
$73$	$ T^{6} + \cdots + 83\!\cdots\!68 $ T^6 + 3553313679653376T^4 + 3639643998132789083472822305280T^2 + 836847664960674275002457788845195789753581568
$79$	$ (T^{3} + \cdots - 17\!\cdots\!68)^{2} $ (T^3 - 93057270T^2 + 2442387979487532T - 17160642935463582495368)^2
$83$	$ T^{6} + \cdots + 50\!\cdots\!32 $ T^6 + 4066673581933440T^4 + 134548705419302065261752325152T^2 + 509185946770108699864239599927389968632832
$89$	$ T^{6} + \cdots + 15\!\cdots\!00 $ T^6 + 10030682450092032T^4 + 2498940007530267394316679897600T^2 + 156937420694458984440438845814793525125120000
$97$	$ T^{6} + \cdots + 12\!\cdots\!28 $ T^6 + 3750067818854400T^4 + 2084462331701819930102708600832T^2 + 122135959248982133284787887516761348561174528
show more
show less

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{5} + ( - \beta_{3} + 361) q^{7}+O(q^{10}) \) q - b1 * q^5 + (-b3 + 361) * q^7	\( q - \beta_1 q^{5} + ( - \beta_{3} + 361) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_1 - 4082) q^{11} + ( - \beta_{4} - 6 \beta_{3} + \cdots - 29 \beta_1) q^{13}+ \cdots + ( - 637 \beta_{4} - 3822 \beta_{3} + \cdots - 28328 \beta_1) q^{97}+O(q^{100}) \) q - b1 * q^5 + (-b3 + 361) * q^7 + (b5 - 2b3 - b1 - 4082) q^11 + (-b4 - 6b3 - 3b2 - 29b1) q^13 + (3b4 + 18b3 + 45b2 - 100b1) * q^17 + (-5b4 - 30b3 + 50b2 - 64b1) * q^19 + (5b5 + 6b4 - 58b3 + 6b2 - 29b1 + 1934) q^23 + (15b5 - 11b4 + 58b3 - 11b2 + 29b1 - 113119) q^25 + (22b5 - 30b4 + 196b3 - 30b2 + 98b1 - 210722) q^29 + (15b4 + 90b3 - 1515b2 + 996b1) * q^31 + (-35b5 - 21b4 - 2b3 - 1596b2 - 189b1 + 219136) q^35 + (30b5 + 78b4 - 684b3 + 78b2 - 342b1 + 530722) q^37 + (-90b4 - 540b3 + 1290b2 - 2078b1) * q^41 + (105b5 - 70b4 + 350b3 - 70b2 + 175b1 - 1297230) q^43 + (81b4 + 486b3 - 4809b2 - 364b1) * q^47 + (-210b5 + 119b4 - 542b3 + 4389b2 - 4172b1 + 453185) q^49 + (-300b5 + 180b4 - 840b3 + 180b2 - 420b1 + 1390110) q^53 + (-375b4 - 2250b3 - 1655b2 + 12948b1) * q^55 + (435b4 + 2610b3 + 5130b2 - 1784b1) * q^59 + (-20b4 - 120b3 + 3300b2 + 10019b1) * q^61 + (35b5 - 483b4 + 3794b3 - 483b2 + 1897b1 - 14015872) q^65 + (-435b5 - 490b4 + 4790b3 - 490b2 + 2395b1 + 2763250) q^67 + (560b5 - 219b4 + 632b3 - 219b2 + 316b1 + 10348142) q^71 + (853b4 + 5118b3 + 33803b2 - 3346b1) * q^73 + (245b5 - 2205b4 + 4418b3 + 28371b2 - 43463b1 + 10231742) q^77 + (834b5 - 471b4 + 2100b3 - 471b2 + 1050b1 + 31019090) q^79 + (-738b4 - 4428b3 - 29163b2 + 43240b1) * q^83 + (2160b5 + 580b4 - 8960b3 + 580b2 - 4480b1 - 43868480) q^85 + (-2445b4 - 14670b3 + 21285b2 - 25554b1) * q^89 + (-2583b5 + 224b4 - 4076b3 - 23401b2 - 39977b1 - 29850176) q^91 + (-2015b5 + 621b4 - 938b3 + 621b2 - 469b1 - 14318816) q^95 + (-637b4 - 3822b3 - 22843b2 - 28328b1) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2166 q^{7}+O(q^{10}) \) 6 * q + 2166 * q^7	\( 6 q + 2166 q^{7} - 24492 q^{11} + 11604 q^{23} - 678714 q^{25} - 1264332 q^{29} + 1314816 q^{35} + 3184332 q^{37} - 7783380 q^{43} + 2719110 q^{49} + 8340660 q^{53} - 84095232 q^{65} + 16579500 q^{67} + 62088852 q^{71} + 61390452 q^{77} + 186114540 q^{79} - 263210880 q^{85} - 179101056 q^{91} - 85912896 q^{95}+O(q^{100}) \) 6 * q + 2166 * q^7 - 24492 * q^11 + 11604 * q^23 - 678714 * q^25 - 1264332 * q^29 + 1314816 * q^35 + 3184332 * q^37 - 7783380 * q^43 + 2719110 * q^49 + 8340660 * q^53 - 84095232 * q^65 + 16579500 * q^67 + 62088852 * q^71 + 61390452 * q^77 + 186114540 * q^79 - 263210880 * q^85 - 179101056 * q^91 - 85912896 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more