LMFDB - Newform orbit 240.6.f.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,6,Mod(49,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$240 = 2^{4} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	240.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:	$38.4921167551$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} + 142x^{6} + 5761x^{4} + 52020x^{2} + 129600$ x^8 + 142x^6 + 5761x^4 + 52020*x^2 + 129600
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{10}\cdot 3^{2}\cdot 5^{3}$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$ .

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$-1$	$1$	$1$

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

	−	2.61388i
		2.11144i
		8.33154i
	−	7.82911i
		2.61388i
	−	2.11144i
	−	8.33154i
		7.82911i

−

9.00000i

−45.4388

32.5625i

−

242.714i

−81.0000

49.2

−

9.00000i

−24.7939

−

50.1025i

27.6635i

−81.0000

49.3

−

9.00000i

−18.3731

52.7961i

122.878i

−81.0000

49.4

−

9.00000i

55.6058

5.74389i

98.1720i

−81.0000

49.5

9.00000i

−45.4388

−

32.5625i

242.714i

−81.0000

49.6

9.00000i

−24.7939

50.1025i

−

27.6635i

−81.0000

49.7

9.00000i

−18.3731

−

52.7961i

−

122.878i

−81.0000

49.8

9.00000i

55.6058

−

5.74389i

−

98.1720i

−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	240.6.f.f		8
3.b	odd	2	1		720.6.f.o		8
4.b	odd	2	1		120.6.f.b	✓	8
5.b	even	2	1	inner	240.6.f.f		8
12.b	even	2	1		360.6.f.c		8
15.d	odd	2	1		720.6.f.o		8
20.d	odd	2	1		120.6.f.b	✓	8
20.e	even	4	1		600.6.a.v		4
20.e	even	4	1		600.6.a.w		4
60.h	even	2	1		360.6.f.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.6.f.b	✓	8	4.b	odd	2	1
120.6.f.b	✓	8	20.d	odd	2	1
240.6.f.f		8	1.a	even	1	1	trivial
240.6.f.f		8	5.b	even	2	1	inner
360.6.f.c		8	12.b	even	2	1
360.6.f.c		8	60.h	even	2	1
600.6.a.v		4	20.e	even	4	1
600.6.a.w		4	20.e	even	4	1
720.6.f.o		8	3.b	odd	2	1
720.6.f.o		8	15.d	odd	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{8} + 84412T_{7}^{6} + 1666776048T_{7}^{4} + 9799162809664T_{7}^{2} + 6560352016000000$ T7^8 + 84412*T7^6 + 1666776048*T7^4 + 9799162809664*T7^2 + 6560352016000000 acting on $S_{6}^{\mathrm{new}}(240, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$(T^{2} + 81)^{4}$ (T^2 + 81)^4
$5$	$T^{8} + \cdots + 95367431640625$ T^8 + 66T^7 + 2460T^6 - 290850T^5 - 22572250T^4 - 908906250T^3 + 24023437500T^2 + 2014160156250*T + 95367431640625
$7$	$T^{8} + \cdots + 65\!\cdots\!00$ T^8 + 84412T^6 + 1666776048T^4 + 9799162809664*T^2 + 6560352016000000
$11$	$(T^{4} + 342 T^{3} + \cdots - 7954368992)^{2}$ (T^4 + 342T^3 - 200492T^2 - 88434840*T - 7954368992)^2
$13$	$T^{8} + \cdots + 28\!\cdots\!64$ T^8 + 3049228T^6 + 3369193592880T^4 + 1610769101740965952*T^2 + 282625364979300524376064
$17$	$T^{8} + \cdots + 18\!\cdots\!96$ T^8 + 7798524T^6 + 15885747884016T^4 + 10278003365875462464*T^2 + 1839527153961828200580096
$19$	$(T^{4} + \cdots - 1452392480768)^{2}$ (T^4 - 1520T^3 - 1886784T^2 + 3787611136*T - 1452392480768)^2
$23$	$T^{8} + \cdots + 91\!\cdots\!00$ T^8 + 14027968T^6 + 70016593387008T^4 + 142612655626816700416*T^2 + 91980808830827677327360000
$29$	$(T^{4} + \cdots - 318720439844864)^{2}$ (T^4 + 4002T^3 - 40798736T^2 - 244204826112*T - 318720439844864)^2
$31$	$(T^{4} + \cdots - 51134602752000)^{2}$ (T^4 + 3012T^3 - 50140800T^2 - 153716486400*T - 51134602752000)^2
$37$	$T^{8} + \cdots + 12\!\cdots\!00$ T^8 + 219152044T^6 + 15254144161777200T^4 + 357929106225129900520000*T^2 + 1281647000497453130012224000000
$41$	$(T^{4} + \cdots + 39\!\cdots\!92)^{2}$ (T^4 + 18900T^3 - 45850064T^2 - 1227533332176*T + 3978546127865392)^2
$43$	$T^{8} + \cdots + 22\!\cdots\!96$ T^8 + 395338176T^6 + 47200836397026816T^4 + 2037527987585392223502336*T^2 + 22604895417105036920358644023296
$47$	$T^{8} + \cdots + 41\!\cdots\!00$ T^8 + 886395984T^6 + 195700091030553600T^4 + 5865002159380182466560000*T^2 + 41468021875648768983957504000000
$53$	$T^{8} + \cdots + 25\!\cdots\!00$ T^8 + 2961585916T^6 + 3214611980604937200T^4 + 1505763282768266463716680000*T^2 + 254236490420860532073913235344000000
$59$	$(T^{4} + \cdots - 69\!\cdots\!32)^{2}$ (T^4 - 43014T^3 + 279226212T^2 + 6744017047320*T - 69521982305008032)^2
$61$	$(T^{4} + \cdots - 15\!\cdots\!28)^{2}$ (T^4 + 19480T^3 - 1559402664T^2 - 38180267839136*T - 157467631166578928)^2
$67$	$T^{8} + \cdots + 34\!\cdots\!24$ T^8 + 10111330768T^6 + 37265322515712015360T^4 + 59475631338290688900984143872*T^2 + 34791456954888200064714801412386586624
$71$	$(T^{4} + \cdots - 12\!\cdots\!32)^{2}$ (T^4 + 81036T^3 + 911625712T^2 - 59365340863680*T - 1218719162661343232)^2
$73$	$T^{8} + \cdots + 35\!\cdots\!16$ T^8 + 4650719344T^6 + 4736033652707602176T^4 + 816382661480960671712333824*T^2 + 35656784074413382646775114180591616
$79$	$(T^{4} + \cdots + 12\!\cdots\!56)^{2}$ (T^4 - 84988T^3 - 6401793696T^2 + 374203814532608*T + 12635467423946899456)^2
$83$	$T^{8} + \cdots + 13\!\cdots\!44$ T^8 + 10279306432T^6 + 32515623935826746880T^4 + 37278824440948265592159059968*T^2 + 13672348730004748759090764892152070144
$89$	$(T^{4} + \cdots - 17\!\cdots\!40)^{2}$ (T^4 + 226488T^3 - 3507369192T^2 - 3436316564346144*T - 177773946658700793840)^2
$97$	$T^{8} + \cdots + 18\!\cdots\!96$ T^8 + 10473438976T^6 + 22444980759068860416T^4 + 14283919293706363847742324736*T^2 + 1803053798426786898957801790684266496
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Classical	Maass
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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

$f(q)$	$=$	$q + 9 \beta_1 q^{3} + (\beta_{2} - 10 \beta_1 - 8) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{7} - 81 q^{9} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \cdots - 86) q^{11} + ( - 5 \beta_{7} + 7 \beta_{5} + \cdots - 5) q^{13}+ \cdots + ( - 162 \beta_{7} - 162 \beta_{6} + \cdots + 6966) q^{99}+O(q^{100})$ q + 9b1 q^3 + (b2 - 10b1 - 8) q^5 + (b5 - b4 - 2b1) q^7 - 81 * q^9 + (2b7 + 2b6 + 3b4 - b3 - 2b2 - b1 - 86) * q^11 + (-5b7 + 7b5 + 4b4 - b3 - 3b2 + 55b1 - 5) q^13 + (9b4 - 72b1 + 90) * q^15 + (-b7 - 9b5 + 6b4 + 5b3 - 11b2 + 157b1 - 1) q^17 + (-5b7 - 9b6 - 8b4 + 2b3 + 2b1 + 379) q^19 + (9b6 + 9b2 + 18) * q^21 + (-2b7 + 3b5 + 12b4 - 11b3 + 20b2 + 193b1 - 2) * q^23 + (-10b7 - 15b6 + 5b5 - 6b4 + 25b3 - 3b2 + 453b1 - 71) q^25 - 729b1 q^27 + (-19b7 - 10b6 - 12b4 + 26b3 + 61b2 + 26b1 - 983) * q^29 + (-3b7 + 21b6 - 42b4 - 36b3 - 48b2 - 36b1 - 759) * q^31 + (9b7 - 18b5 - 18b4 + 18b3 - 27b2 - 774b1 + 9) * q^33 + (10b7 - 40b6 - 45b5 - 27b4 + 10b3 - 10b2 - 3374b1 + 570) q^35 + (-29b7 - 9b5 + 44b4 + 23b3 - 75b2 + 3543b1 - 29) * q^37 + (9b7 + 63b6 - 27b4 - 45b3 - 36b2 - 45b1 - 495) * q^39 + (-48b7 + 14b6 - 96b4 + 62b2 - 4694) * q^41 + (-24b7 + 18b5 + 84b4 - 54b3 + 84b2 + 2886b1 - 24) * q^43 + (-81b2 + 810b1 + 648) * q^45 + (14b7 + 39b5 + 18b4 - 85b3 + 184b2 - 1877b1 + 14) * q^47 + (-48b7 + 138b6 - 132b4 - 36b3 + 114b2 - 36b1 - 4221) * q^49 + (-45b7 - 81b6 - 99b4 - 9b3 - 54b2 - 9b1 - 1413) * q^51 + (-161b7 + 78b5 + 312b4 - 68b3 - 25b2 - 11768b1 - 161) * q^53 + (-55b7 - 15b6 + 105b5 - 5b4 - 110b3 - 132b2 - 8100b1 - 5099) q^55 + (-18b7 + 81b5 - 45b3 + 72b2 + 3411b1 - 18) q^57 + (16b7 + 116b6 + 9b4 - 23b3 + 54b2 - 23b1 + 10792) * q^59 + (-32b7 - 216b6 - 56b4 + 8b3 - 168b2 + 8b1 - 4958) * q^61 + (-81b5 + 81b4 + 162b1) q^63 + (-175b7 - 180b6 - 165b5 - 18b4 - 115b3 + 5b2 + 7129b1 + 10455) q^65 + (-176b7 - 204b5 + 470b4 + 86b3 - 348b2 - 6258b1 - 176) * q^67 + (99b7 + 27b6 + 180b4 - 18b3 - 108b2 - 18b1 - 1737) * q^69 + (2b7 - 214b6 + 54b4 + 50b3 - 116b2 + 50b1 - 20342) * q^71 + (86b7 - 160b5 - 16b4 + 4b3 + 78b2 - 27796b1 + 86) * q^73 + (-225b7 + 45b6 + 135b5 - 27b4 - 90b3 + 54b2 - 639b1 - 4077) q^75 + (-130b7 + 108b5 + 372b4 - 220b3 + 310b2 - 11316b1 - 130) * q^77 + (-395b7 - 75b6 - 770b4 + 20b3 + 360b2 + 20b1 + 21417) * q^79 + 6561 * q^81 + (-220b7 + 228b5 + 84b4 + 128b3 - 476b2 + 20732b1 - 220) * q^83 + (95b7 + 510b6 + 305b5 + 184b4 - 335b3 + 105b2 + 17293b1 + 30045) q^85 + (-234b7 + 90b5 + 549b4 - 171b3 + 108b2 - 8847b1 - 234) * q^87 + (-294b7 + 162b6 - 1200b4 - 612b3 - 768b2 - 612b1 - 56700) * q^89 + (-398b7 + 402b6 - 578b4 + 218b3 + 1236b2 + 218b1 - 75798) * q^91 + (324b7 - 189b5 - 432b4 - 27b3 + 378b2 - 6831b1 + 324) * q^93 + (200b7 + 230b6 - 435b5 + 60b4 + 215b3 + 388b2 + 17455b1 - 4904) q^95 + (-76b7 - 292b5 + 452b4 - 8b3 - 60b2 - 7816b1 - 76) * q^97 + (-162b7 - 162b6 - 243b4 + 81b3 + 162b2 + 81b1 + 6966) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 66 q^{5} - 648 q^{9} - 684 q^{11} + 738 q^{15} + 3040 q^{19} + 108 q^{21} - 564 q^{25} - 8004 q^{29} - 6024 q^{31} + 4476 q^{35} - 3996 q^{39} - 37800 q^{41} + 5346 q^{45} - 34368 q^{49} - 11124 q^{51}+ \cdots + 55404 q^{99}+O(q^{100})$ 8 * q - 66 * q^5 - 648 * q^9 - 684 * q^11 + 738 * q^15 + 3040 * q^19 + 108 * q^21 - 564 * q^25 - 8004 * q^29 - 6024 * q^31 + 4476 * q^35 - 3996 * q^39 - 37800 * q^41 + 5346 * q^45 - 34368 * q^49 - 11124 * q^51 - 39968 * q^55 + 86028 * q^59 - 38960 * q^61 + 84204 * q^65 - 13536 * q^69 - 162072 * q^71 - 31968 * q^75 + 169976 * q^79 + 52488 * q^81 + 240588 * q^85 - 452976 * q^89 - 610456 * q^91 - 42048 * q^95 + 55404 * q^99

$\beta_{1}$	$=$	$( -\nu^{7} - 142\nu^{5} - 5401\nu^{3} - 26460\nu ) / 10800$ (-v^7 - 142v^5 - 5401v^3 - 26460*v) / 10800
$\beta_{2}$	$=$	$( -5\nu^{7} + 12\nu^{6} - 674\nu^{5} + 1272\nu^{4} - 25169\nu^{3} + 31980\nu^{2} - 160740\nu + 82080 ) / 2160$ (-5v^7 + 12v^6 - 674v^5 + 1272v^4 - 25169v^3 + 31980v^2 - 160740*v + 82080) / 2160
$\beta_{3}$	$=$	$( 19\nu^{7} + 2698\nu^{5} - 2700\nu^{4} + 111619\nu^{3} - 218700\nu^{2} + 979740\nu - 1933200 ) / 10800$ (19v^7 + 2698v^5 - 2700v^4 + 111619v^3 - 218700v^2 + 979740v - 1933200) / 10800
$\beta_{4}$	$=$	$( 8\nu^{7} - 30\nu^{6} + 1046\nu^{5} - 3855\nu^{4} + 36368\nu^{3} - 134625\nu^{2} + 163530\nu - 688500 ) / 2700$ (8v^7 - 30v^6 + 1046v^5 - 3855v^4 + 36368v^3 - 134625v^2 + 163530*v - 688500) / 2700
$\beta_{5}$	$=$	$( -22\nu^{7} - 30\nu^{6} - 2854\nu^{5} - 3855\nu^{4} - 97402\nu^{3} - 134625\nu^{2} - 325170\nu - 688500 ) / 2700$ (-22v^7 - 30v^6 - 2854v^5 - 3855v^4 - 97402v^3 - 134625v^2 - 325170*v - 688500) / 2700
$\beta_{6}$	$=$	$( 5\nu^{7} + 60\nu^{6} + 674\nu^{5} + 8520\nu^{4} + 25169\nu^{3} + 317580\nu^{2} + 160740\nu + 1343520 ) / 2160$ (5v^7 + 60v^6 + 674v^5 + 8520v^4 + 25169v^3 + 317580v^2 + 160740*v + 1343520) / 2160
$\beta_{7}$	$=$	$( - 89 \nu^{7} - 300 \nu^{6} - 11738 \nu^{5} - 37200 \nu^{4} - 416789 \nu^{3} - 1236900 \nu^{2} + \cdots - 5929200 ) / 10800$ (-89v^7 - 300v^6 - 11738v^5 - 37200v^4 - 416789v^3 - 1236900v^2 - 2111940*v - 5929200) / 10800

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	240.6.f.f

Level	$240$
Weight	$6$
Character orbit	240.f
Analytic conductor	$38.492$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

$\nu$	$=$	$( -4\beta_{7} + 5\beta_{5} + 6\beta_{4} - 3\beta_{3} + 2\beta_{2} + \beta _1 - 4 ) / 120$ (-4b7 + 5b5 + 6b4 - 3b3 + 2*b2 + b1 - 4) / 120
$\nu^{2}$	$=$	$( -3\beta_{7} - 5\beta_{6} - 10\beta_{4} - 4\beta_{3} - 10\beta_{2} - 4\beta _1 - 1423 ) / 40$ (-3b7 - 5b6 - 10b4 - 4b3 - 10b2 - 4b1 - 1423) / 40
$\nu^{3}$	$=$	$( 212\beta_{7} - 265\beta_{5} - 390\beta_{4} + 231\beta_{3} - 250\beta_{2} + 2611\beta _1 + 212 ) / 120$ (212b7 - 265b5 - 390b4 + 231b3 - 250b2 + 2611b1 + 212) / 120
$\nu^{4}$	$=$	$( 243\beta_{7} + 405\beta_{6} + 730\beta_{4} + 244\beta_{3} + 650\beta_{2} + 244\beta _1 + 86623 ) / 40$ (243b7 + 405b6 + 730b4 + 244b3 + 650b2 + 244b1 + 86623) / 40
$\nu^{5}$	$=$	$( -13252\beta_{7} + 17465\beta_{5} + 24630\beta_{4} - 15591\beta_{3} + 17930\beta_{2} - 313811\beta _1 - 13252 ) / 120$ (-13252b7 + 17465b5 + 24630b4 - 15591b3 + 17930b2 - 313811b1 - 13252) / 120
$\nu^{6}$	$=$	$( -18483\beta_{7} - 29605\beta_{6} - 50730\beta_{4} - 13764\beta_{3} - 38650\beta_{2} - 13764\beta _1 - 5664063 ) / 40$ (-18483b7 - 29605b6 - 50730b4 - 13764b3 - 38650b2 - 13764b1 - 5664063) / 40
$\nu^{7}$	$=$	$( 842612 \beta_{7} - 1181065 \beta_{5} - 1549830 \beta_{4} + 1045671 \beta_{3} - 1248730 \beta_{2} + \cdots + 842612 ) / 120$ (842612b7 - 1181065b5 - 1549830b4 + 1045671b3 - 1248730b2 + 29136691b1 + 842612) / 120

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.8
Significant digits:
Format: