LMFDB - Newform orbit 252.5.z.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,5,Mod(73,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.73"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$N$	$=$	$252 = 2^{2} \cdot 3^{2} \cdot 7$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	252.z (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,27,0,66] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$26.0492306971$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.11337408.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} + 18x^{4} + 81x^{2} + 12$ x^6 + 18x^4 + 81x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{2}\cdot 3^{3}\cdot 7^{2}$
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Character values

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

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$1 - \beta_{1}$	$1$

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}

73.1

	−	0.391571i
	−	2.78499i
		3.17656i
		0.391571i
		2.78499i
	−	3.17656i

−38.3327

22.1314i

42.3967

24.5667i

73.2

24.9067

−

14.3799i

39.2754

−

29.2993i

73.3

26.9260

−

15.5457i

−48.6720

−

5.65972i

145.1

−38.3327

−

22.1314i

42.3967

−

24.5667i

145.2

24.9067

14.3799i

39.2754

29.2993i

145.3

26.9260

15.5457i

−48.6720

5.65972i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.5.z.f		6
3.b	odd	2	1		28.5.h.a	✓	6
7.d	odd	6	1	inner	252.5.z.f		6
12.b	even	2	1		112.5.s.c		6
15.d	odd	2	1		700.5.s.a		6
15.e	even	4	2		700.5.o.a		12
21.c	even	2	1		196.5.h.c		6
21.g	even	6	1		28.5.h.a	✓	6
21.g	even	6	1		196.5.b.a		6
21.h	odd	6	1		196.5.b.a		6
21.h	odd	6	1		196.5.h.c		6
84.j	odd	6	1		112.5.s.c		6
84.j	odd	6	1		784.5.c.e		6
84.n	even	6	1		784.5.c.e		6
105.p	even	6	1		700.5.s.a		6
105.w	odd	12	2		700.5.o.a		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.5.h.a	✓	6	3.b	odd	2	1
28.5.h.a	✓	6	21.g	even	6	1
112.5.s.c		6	12.b	even	2	1
112.5.s.c		6	84.j	odd	6	1
196.5.b.a		6	21.g	even	6	1
196.5.b.a		6	21.h	odd	6	1
196.5.h.c		6	21.c	even	2	1
196.5.h.c		6	21.h	odd	6	1
252.5.z.f		6	1.a	even	1	1	trivial
252.5.z.f		6	7.d	odd	6	1	inner
700.5.o.a		12	15.e	even	4	2
700.5.o.a		12	105.w	odd	12	2
700.5.s.a		6	15.d	odd	2	1
700.5.s.a		6	105.p	even	6	1
784.5.c.e		6	84.j	odd	6	1
784.5.c.e		6	84.n	even	6	1

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{6} - 27T_{5}^{5} - 1512T_{5}^{4} + 47385T_{5}^{3} + 2463048T_{5}^{2} - 120310515T_{5} + 1566504603$ T5^6 - 27*T5^5 - 1512*T5^4 + 47385*T5^3 + 2463048*T5^2 - 120310515*T5 + 1566504603 acting on $S_{5}^{\mathrm{new}}(252, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6}$ T^6
$5$	$T^{6} + \cdots + 1566504603$ T^6 - 27T^5 - 1512T^4 + 47385T^3 + 2463048T^2 - 120310515*T + 1566504603
$7$	$T^{6} + \cdots + 13841287201$ T^6 - 66T^5 - 2037T^4 + 331436T^3 - 4890837T^2 - 380476866*T + 13841287201
$11$	$T^{6} + \cdots + 1920280041$ T^6 + 135T^5 + 20088T^4 - 163863T^3 + 9386604T^2 + 81638523*T + 1920280041
$13$	$T^{6} + \cdots + 65933832204288$ T^6 + 143208T^4 + 5865137424T^2 + 65933832204288
$17$	$T^{6} + \cdots + 86645980184427$ T^6 - 1107T^5 + 542808T^4 - 148697775T^3 + 23992441704T^2 - 2165667036075*T + 86645980184427
$19$	$T^{6} + \cdots + 163848759776883$ T^6 + 747T^5 + 150318T^4 - 26656695T^3 - 4247120682T^2 + 791166532455*T + 163848759776883
$23$	$T^{6} + \cdots + 18\!\cdots\!21$ T^6 + 243T^5 + 380700T^4 - 164398815T^3 + 92981494728T^2 - 13869208676961*T + 1859231862053721
$29$	$(T^{3} - 270 T^{2} + \cdots - 401089968)^{2}$ (T^3 - 270T^2 - 1261656T - 401089968)^2
$31$	$T^{6} + \cdots + 81\!\cdots\!63$ T^6 + 5355T^5 + 12530574T^4 + 15914519145T^3 + 11629710845046T^2 + 4657685280270183*T + 818749284432880563
$37$	$T^{6} + \cdots + 19\!\cdots\!81$ T^6 - 2355T^5 + 5983494T^4 - 1762232087T^3 + 3479514413766T^2 - 610809875252979*T + 1949474384069395681
$41$	$T^{6} + \cdots + 30\!\cdots\!92$ T^6 + 14908968T^4 + 58812570389520T^2 + 30617085863456980992
$43$	$(T^{3} + 474 T^{2} + \cdots + 2925826856)^{2}$ (T^3 + 474T^2 - 4592652T + 2925826856)^2
$47$	$T^{6} + \cdots + 12\!\cdots\!27$ T^6 - 9747T^5 + 40165902T^4 - 82829021553T^3 + 91714282823142T^2 - 51003075249505791*T + 12007382371055101827
$53$	$T^{6} + \cdots + 34\!\cdots\!01$ T^6 + 6291T^5 + 31845798T^4 + 52357433751T^3 + 71475514653798T^2 - 14388908065414317*T + 3464156263432911201
$59$	$T^{6} + \cdots + 40\!\cdots\!87$ T^6 - 2943T^5 - 13578894T^4 + 48459370311T^3 + 237132183168090T^2 - 570622732726616163*T + 400315010438243761587
$61$	$T^{6} + \cdots + 15\!\cdots\!47$ T^6 - 4041T^5 - 82908T^4 + 22331111535T^3 + 1270138682988T^2 - 120073557090573585*T + 157373106493448603547
$67$	$T^{6} + \cdots + 30\!\cdots\!41$ T^6 - 1659T^5 + 46628988T^4 + 83814690071T^3 + 1916021645259288T^2 + 241831585733125353*T + 30377917313907663241
$71$	$(T^{3} + 1134 T^{2} + \cdots - 66080643048)^{2}$ (T^3 + 1134T^2 - 57518748T - 66080643048)^2
$73$	$T^{6} + \cdots + 51\!\cdots\!67$ T^6 + 8703T^5 + 10727892T^4 - 126363304233T^3 - 149632437632580T^2 + 1804046175243796311*T + 5146003339031035005867
$79$	$T^{6} + \cdots + 32\!\cdots\!81$ T^6 + 7773T^5 + 67927740T^4 + 56054451215T^3 + 501050143193928T^2 + 429528891408068049*T + 3272742410404764500281
$83$	$T^{6} + \cdots + 21\!\cdots\!72$ T^6 + 73570464T^4 + 772925219778816T^2 + 2119841468956608233472
$89$	$T^{6} + \cdots + 12\!\cdots\!07$ T^6 + 14985T^5 + 31277988T^4 - 652927723695T^3 - 1124094141465876T^2 + 26366746972549723857*T + 122060494227189993108507
$97$	$T^{6} + \cdots + 22\!\cdots\!32$ T^6 + 161932968T^4 + 6209172331595280T^2 + 222775495464660959232
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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

$f(q)$	$=$	$q + ( - \beta_{4} + \beta_{2} + 3 \beta_1 + 3) q^{5} + ( - \beta_{5} + 3 \beta_{2} + 4 \beta_1 + 9) q^{7} + (\beta_{5} + 2 \beta_{4} + \cdots - 45 \beta_1) q^{11} + ( - \beta_{5} - \beta_{4} - 21 \beta_{3} + \cdots + 12) q^{13}+ \cdots + (59 \beta_{5} + 59 \beta_{4} + \cdots - 3476) q^{97}+O(q^{100})$ q + (-b4 + b2 + 3b1 + 3) q^5 + (-b5 + 3b2 + 4b1 + 9) * q^7 + (b5 + 2b4 - 2b3 + 4b2 - 45b1) * q^11 + (-b5 - b4 - 21b3 + 21b2 - 24b1 + 12) q^13 + (-b5 - b3 - 123b1 + 246) q^17 + (-7b4 - 6b2 - 83b1 - 83) q^19 + (16b5 + 8b4 + 10b3 - 5b2 + 81b1 - 81) q^23 + (8b5 + 16b4 + 12b3 - 24b2 + 626b1) q^25 + (9b5 - 9b4 - 39b3 - 39b2 + 90) * q^29 + (-9b5 + 18b3 + 595b1 - 1190) q^31 + (11b5 - 10b4 - 19b3 + 133b2 + 267b1 - 1137) q^35 + (40b5 + 20b4 - 24b3 + 12b2 - 785b1 + 785) q^37 + (29b5 + 29b4 - 115b3 + 115b2 + 1752b1 - 876) q^41 + (28b5 - 28b4 + 42b3 + 42b2 - 158) * q^43 + (-19b4 + 100b2 + 1083b1 + 1083) q^47 + (-27b5 - 37b4 - 183b3 + 117b2 - 128b1 + 1469) q^49 + (8b5 + 16b4 - 100b3 + 200b2 - 2097b1) q^53 + (30b5 + 30b4 - 135b3 + 135b2 - 2286b1 + 1143) q^55 + (10b5 - 323b3 - 327b1 + 654) q^59 + (-64b4 + 48b2 + 449b1 + 449) q^61 + (126b5 + 63b4 - 546b3 + 273b2 + 2556b1 - 2556) q^65 + (90b5 + 180b4 - 33b3 + 66b2 + 553b1) q^67 + (70b5 - 70b4 + 280b3 + 280b2 - 378) * q^71 + (-106b5 - 174b3 + 967b1 - 1934) q^73 + (-28b5 - 59b4 + 128b3 - 289b2 + 2187b1 + 684) q^77 + (104b5 + 52b4 + 366b3 - 183b2 + 2591b1 - 2591) q^79 + (86b5 + 86b4 + 302b3 - 302b2 - 3216b1 + 1608) q^83 + (134b5 - 134b4 + 138b3 + 138b2 - 117) * q^85 + (-96b4 + 600b2 - 1665b1 - 1665) q^89 + (-205b5 - 231b4 - 483b3 + 69b2 + 6000b1 - 6492) q^91 + (134b5 + 268b4 + 383b3 - 766b2 + 6885b1) q^95 + (59b5 + 59b4 + 375b3 - 375b2 + 6952b1 - 3476) q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 27 q^{5} + 66 q^{7} - 135 q^{11} + 1107 q^{17} - 747 q^{19} - 243 q^{23} + 1878 q^{25} + 540 q^{29} - 5355 q^{31} - 6021 q^{35} + 2355 q^{37} - 948 q^{43} + 9747 q^{47} + 8430 q^{49} - 6291 q^{53}+ \cdots + 20655 q^{95}+O(q^{100})$ 6 * q + 27 * q^5 + 66 * q^7 - 135 * q^11 + 1107 * q^17 - 747 * q^19 - 243 * q^23 + 1878 * q^25 + 540 * q^29 - 5355 * q^31 - 6021 * q^35 + 2355 * q^37 - 948 * q^43 + 9747 * q^47 + 8430 * q^49 - 6291 * q^53 + 2943 * q^59 + 4041 * q^61 - 7668 * q^65 + 1659 * q^67 - 2268 * q^71 - 8703 * q^73 + 10665 * q^77 - 7773 * q^79 - 702 * q^85 - 14985 * q^89 - 20952 * q^91 + 20655 * q^95

$\beta_{1}$	$=$	$( \nu^{3} + 9\nu + 2 ) / 4$ (v^3 + 9*v + 2) / 4
$\beta_{2}$	$=$	$( -\nu^{5} - 9\nu^{4} - 15\nu^{3} - 87\nu^{2} - 36\nu - 36 ) / 8$ (-v^5 - 9v^4 - 15v^3 - 87v^2 - 36v - 36) / 8
$\beta_{3}$	$=$	$( \nu^{5} - 9\nu^{4} + 15\nu^{3} - 87\nu^{2} + 36\nu - 36 ) / 8$ (v^5 - 9v^4 + 15v^3 - 87v^2 + 36v - 36) / 8
$\beta_{4}$	$=$	$( 9\nu^{5} - 3\nu^{4} + 135\nu^{3} + 27\nu^{2} + 492\nu + 324 ) / 8$ (9v^5 - 3v^4 + 135v^3 + 27v^2 + 492*v + 324) / 8
$\beta_{5}$	$=$	$( 9\nu^{5} + 3\nu^{4} + 135\nu^{3} - 27\nu^{2} + 492\nu - 324 ) / 8$ (9v^5 + 3v^4 + 135v^3 - 27v^2 + 492*v - 324) / 8

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