LMFDB - Newform orbit 960.4.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(289,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 $ x^12 - 6x^11 + 18x^10 + 300x^9 + 14598x^8 - 60928x^7 + 147804x^6 + 1450180x^5 + 13518601x^4 - 29643466x^3 + 68047778x^2 + 610365120*x + 2737382400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 3 q^{3} - \beta_{4} q^{5} + \beta_{6} q^{7} + 9 q^{9} + ( - \beta_{11} - 2 \beta_{3}) q^{11} + ( - \beta_{10} - \beta_{5}) q^{13} + 3 \beta_{4} q^{15} + (\beta_{11} - \beta_{9} + 5 \beta_{3}) q^{17}+ \cdots + ( - 9 \beta_{11} - 18 \beta_{3}) q^{99}+O(q^{100}) $ q - 3 * q^3 - b4 * q^5 + b6 * q^7 + 9 * q^9 + (-b11 - 2b3) q^11 + (-b10 - b5) * q^13 + 3b4 q^15 + (b11 - b9 + 5b3) q^17 + (-2b11 - b9 + 2b3) * q^19 - 3b6 q^21 + (2b8 + 5b7 - 3b6 - 5b4) * q^23 + (-2b11 - b9 - 2b3 + b2 + b1 - 4) * q^25 - 27 * q^27 + (-3b8 - 11b7 + 6b6 + 11b4) * q^29 + (2b10 + 3b7 - 3b5 + 3b4) * q^31 + (3b11 + 6b3) * q^33 + (-2b9 + 14b3 - 2b2 + b1 - 24) q^35 + (-6b7 + 5b5 - 6b4) q^37 + (3b10 + 3b5) * q^39 + (-b3 - 3b2 - 2b1 + 67) * q^41 + (-2b3 - b2 - 4b1 + 123) * q^43 - 9b4 q^45 + (3b8 + 9b7 + 3b6 - 9b4) * q^47 + (2b3 + b2 + 4b1 - 124) * q^49 + (-3b11 + 3b9 - 15b3) q^51 + (-5b10 - 9b7 - 9b4) q^53 + (7b10 + 3b8 + 15b7 - 5b6 + 15b5 - 9b4) * q^55 + (6b11 + 3b9 - 6b3) q^57 + (11b11 + 2b9 - 41b3) q^59 + (-9b8 - 4b7 - 18b6 + 4b4) * q^61 + 9b6 q^63 + (-7b11 + b9 - 56b3 - 2b2 + 80) q^65 + (9b2 - 99) q^67 + (-6b8 - 15b7 + 9b6 + 15b4) * q^69 + (6b10 + 30b7 - 30b5 + 30b4) * q^71 + (-12b11 + 6b9 + 9b3) q^73 + (6b11 + 3b9 + 6b3 - 3b2 - 3b1 + 12) q^75 + (-27b10 + 2b7 - 12b5 + 2b4) * q^77 + (-20b10 + 45b7 - 9b5 + 45b4) * q^79 + 81 * q^81 + (5b3 - 2b2 + 10b1 + 102) q^83 + (-20b10 - 15b8 + 10b7 + 30b6 - 15b5 + 20b4) * q^85 + (9b8 + 33b7 - 18b6 - 33b4) * q^87 + (-7b3 - b2 - 14b1 - 259) * q^89 + (18b11 + 86b3) * q^91 + (-6b10 - 9b7 + 9b5 - 9b4) * q^93 + (-2b10 - 3b8 + 55b7 + 15b6 + 30b5 - 7b4) * q^95 + (14b11 - 2b9 + 94b3) q^97 + (-9b11 - 18b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{3} + 108 q^{9} - 44 q^{25} - 324 q^{27} - 296 q^{35} + 792 q^{41} + 1472 q^{43} - 1484 q^{49} + 952 q^{65} - 1152 q^{67} + 132 q^{75} + 972 q^{81} + 1216 q^{83} - 3112 q^{89}+O(q^{100}) $ 12 * q - 36 * q^3 + 108 * q^9 - 44 * q^25 - 324 * q^27 - 296 * q^35 + 792 * q^41 + 1472 * q^43 - 1484 * q^49 + 952 * q^65 - 1152 * q^67 + 132 * q^75 + 972 * q^81 + 1216 * q^83 - 3112 * q^89

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 $ x^12 - 6*x^11 + 18*x^10 + 300*x^9 + 14598*x^8 - 60928*x^7 + 147804*x^6 + 1450180*x^5 + 13518601*x^4 - 29643466*x^3 + 68047778*x^2 + 610365120*x + 2737382400 :

$\beta_{1}$	$=$	$ ( - 28606200764657 \nu^{11} - 107678553130919 \nu^{10} + \cdots + 85\!\cdots\!00 ) / 53\!\cdots\!00 $ (-28606200764657v^11 - 107678553130919v^10 + 14286839377994035v^9 - 324451897654674005v^8 + 639265766735073529v^7 - 183244140882525627v^6 + 124847626693067751525v^5 - 3709540294441962096995v^4 + 6559956283742582410928v^3 + 4922688670902430074346v^2 - 78070757530701486089760*v + 853114019950910061388800) / 53240300062167380256000
$\beta_{2}$	$=$	$ ( 5249469155 \nu^{11} - 145077666139 \nu^{10} + 3094091359655 \nu^{9} - 52134431764465 \nu^{8} + \cdots - 45\!\cdots\!00 ) / 30\!\cdots\!00 $ (5249469155v^11 - 145077666139v^10 + 3094091359655v^9 - 52134431764465v^8 + 232855611868085v^7 - 1618624475819007v^6 + 30133784567176305v^5 - 724640028919396135v^4 + 1367729435430712960v^3 - 787680124180438654v^2 - 13142984232319037760*v - 453190011628080643200) / 3015080986644432000
$\beta_{3}$	$=$	$ ( 6915331370263 \nu^{11} - 80794468014023 \nu^{10} + 323442252748465 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 33\!\cdots\!00 $ (6915331370263v^11 - 80794468014023v^10 + 323442252748465v^9 + 1500264010761355v^8 + 89890347406849759v^7 - 1024615208733730479v^6 + 2900643934037276355v^5 + 5359108108690675045v^4 + 46570369016402753978v^3 - 1124621592758331401498v^2 + 1358902948610911125180*v + 2074742688462266721600) / 3327518753885461266000
$\beta_{4}$	$=$	$ ( 40\!\cdots\!51 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 70\!\cdots\!00 $ (40754795208499751v^11 - 485566357915119265v^10 + 5023378005722421635v^9 - 5353665270106381255v^8 + 562982360221268471213v^7 - 5135027942476227647205v^6 + 61875184181441159964765v^5 - 86826217107605079155545v^4 + 610941866986855714064236v^3 - 783402517961408900914330v^2 + 31555673942476677728776800*v + 3007772859969465191712000) / 7080959908268261574048000
$\beta_{5}$	$=$	$ ( - 11\!\cdots\!19 \nu^{11} + \cdots - 78\!\cdots\!00 ) / 64\!\cdots\!00 $ (-1179207355312919v^11 + 5055403466864107v^10 + 2250309511779445v^9 - 435564475626491735v^8 - 17724183040865622857v^7 + 45608956902906922431v^6 + 135625619458107816675v^5 - 2403707765863503043265v^4 - 16980502844995568071624v^3 + 21371140519339045256062v^2 + 238637388409374724956480*v - 783064564813946719526400) / 64962934938240931872000
$\beta_{6}$	$=$	$ ( 19\!\cdots\!93 \nu^{11} + \cdots + 72\!\cdots\!00 ) / 87\!\cdots\!00 $ (1934372922803593v^11 - 12716860284760433v^10 + 59114914395710965v^9 + 473792237723756605v^8 + 28165174889944007599v^7 - 129253959831677193309v^6 + 635931766652094328755v^5 + 1833658999221919156795v^4 + 25037631342969820921808v^3 - 47264922583194723659258v^2 + 529212600963997979429280*v + 724291611935477269593600) / 87419258126768661408000
$\beta_{7}$	$=$	$ ( - 16\!\cdots\!19 \nu^{11} + \cdots - 66\!\cdots\!00 ) / 70\!\cdots\!00 $ (-16556965419423019v^11 + 92306962583284286v^10 - 254100039576323155v^9 - 5318511198810265000v^8 - 236425806837572067127v^7 + 882342486613512256698v^6 - 2016969655917771927105v^5 - 26448337452075259946500v^4 - 131917787806858289245154v^3 + 248686201635138574053116v^2 - 1629999844090684794607440*v - 6644713394874573633225600) / 708095990826826157404800
$\beta_{8}$	$=$	$ ( 215567487415101 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 48\!\cdots\!00 $ (215567487415101v^11 - 1523965074430661v^10 + 9066039751198105v^9 + 47280778653465985v^8 + 3003716617850958043v^7 - 14910159110254833953v^6 + 101495732455616332335v^5 + 167403641542706185815v^4 + 1237428709276880787856v^3 - 1547989058908558208386v^2 + 41103419556176793782560*v + 64617229350542493427200) / 4856625451487147856000
$\beta_{9}$	$=$	$ ( - 12\!\cdots\!61 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 26\!\cdots\!00 $ (-1208724988289461v^11 + 18551517427273769v^10 - 78057188646759865v^9 - 222065911156639405v^8 - 13890064213629217363v^7 + 232139909686647025557v^6 - 668068950346366078095v^5 - 681606620985336775195v^4 + 1567425413914970049424v^3 + 78970772770666714537274v^2 - 120421318072287770131440*v - 252396365944789055980800) / 26620150031083690128000
$\beta_{10}$	$=$	$ ( - 169860038210899 \nu^{11} + 661777164565559 \nu^{10} + \cdots - 99\!\cdots\!00 ) / 36\!\cdots\!00 $ (-169860038210899v^11 + 661777164565559v^10 + 1907527771683305v^9 - 71667946412210515v^8 - 2490482561363103757v^7 + 5797301894803039707v^6 + 37448808346181987535v^5 - 408998372404918971685v^4 - 1719387721142576106344v^3 + 2699941523000578935734v^2 + 27659641534183733444160*v - 99635781138591912652800) / 3609051941013385104000
$\beta_{11}$	$=$	$ ( 26\!\cdots\!45 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 53\!\cdots\!00 $ (2628210621539645v^11 - 36176790982136089v^10 + 149504680672139705v^9 + 519887465865230285v^8 + 31971951826365576755v^7 - 458238056093189879157v^6 + 1310738298755515797855v^5 + 1715822323522946653115v^4 + 5372109912204030857800v^3 - 278590875798475619024554v^2 + 368426352800655831596640*v + 649718073381682590220800) / 53240300062167380256000

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

$\nu$	$=$	$ ( \beta_{10} - \beta_{8} - 4\beta_{7} - \beta_{3} + 4 ) / 8 $ (b10 - b8 - 4*b7 - b3 + 4) / 8
$\nu^{2}$	$=$	$ ( 4\beta_{11} + 2\beta_{9} - \beta_{8} - 2\beta_{7} + 2\beta_{4} - 58\beta_{3} ) / 4 $ (4b11 + 2b9 - b8 - 2b7 + 2b4 - 58*b3) / 4
$\nu^{3}$	$=$	$ ( 12 \beta_{11} - 90 \beta_{10} + 6 \beta_{9} - 73 \beta_{8} - 34 \beta_{7} - 90 \beta_{6} + 90 \beta_{5} + \cdots - 671 ) / 8 $ (12b11 - 90b10 + 6b9 - 73b8 - 34b7 - 90b6 + 90b5 + 326b4 - 170b3 - 3b2 + 6*b1 - 671) / 8
$\nu^{4}$	$=$	$ ( -178\beta_{10} + 288\beta_{7} + 180\beta_{5} + 288\beta_{4} + 159\beta_{3} - 111\beta_{2} + 318\beta _1 - 22043 ) / 4 $ (-178b10 + 288b7 + 180b5 + 288b4 + 159b3 - 111b2 + 318*b1 - 22043) / 4
$\nu^{5}$	$=$	$ ( - 2660 \beta_{11} - 10764 \beta_{10} - 1570 \beta_{9} + 7499 \beta_{8} + 36526 \beta_{7} + \cdots - 107981 ) / 8 $ (-2660b11 - 10764b10 - 1570b9 + 7499b8 + 36526b7 + 13530b6 + 13530b5 - 6530b4 + 28974b3 - 545b2 + 1570*b1 - 107981) / 8
$\nu^{6}$	$=$	$ ( - 68416 \beta_{11} - 42608 \beta_{9} + 21779 \beta_{8} + 62808 \beta_{7} + 39690 \beta_{6} + \cdots + 687052 \beta_{3} ) / 4 $ (-68416b11 - 42608b9 + 21779b8 + 62808b7 + 39690b6 - 62808b4 + 687052*b3) / 4
$\nu^{7}$	$=$	$ ( - 460348 \beta_{11} + 1362080 \beta_{10} - 287294 \beta_{9} + 897887 \beta_{8} + 928386 \beta_{7} + \cdots + 17617219 ) / 8 $ (-460348b11 + 1362080b10 - 287294b9 + 897887b8 + 928386b7 + 1789050b6 - 1789050b5 - 4519934b4 + 4469200b3 + 86527b2 - 287294*b1 + 17617219) / 8
$\nu^{8}$	$=$	$ ( 5156872 \beta_{10} - 6779232 \beta_{7} - 6787440 \beta_{5} - 6779232 \beta_{4} - 2775691 \beta_{3} + \cdots + 331530127 ) / 4 $ (5156872b10 - 6779232b7 - 6787440b5 - 6779232b4 - 2775691b3 + 1603579b2 - 5551382*b1 + 331530127) / 4
$\nu^{9}$	$=$	$ ( 73347180 \beta_{11} + 175719976 \beta_{10} + 46541190 \beta_{9} - 113872741 \beta_{8} + \cdots + 2774167879 ) / 8 $ (73347180b11 + 175719976b10 + 46541190b9 - 113872741b8 - 579185434b7 - 233157390b6 - 233157390b5 + 123694470b4 - 750135406b3 + 13402995b2 - 46541190*b1 + 2774167879) / 8
$\nu^{10}$	$=$	$ ( 1136905264 \beta_{11} + 724446632 \beta_{9} - 519123541 \beta_{8} - 1604238032 \beta_{7} + \cdots - 11242050388 \beta_{3} ) / 4 $ (1136905264b11 + 724446632b9 - 519123541b8 - 1604238032b7 - 1065079470b6 + 1604238032b4 - 11242050388*b3) / 4
$\nu^{11}$	$=$	$ ( 11181281892 \beta_{11} - 22961104200 \beta_{10} + 7128161106 \beta_{9} - 14804378353 \beta_{8} + \cdots - 421812793541 ) / 8 $ (11181281892b11 - 22961104200b10 + 7128161106b9 - 14804378353b8 - 16313451694b7 - 30545297910b6 + 30545297910b5 + 75530965106b4 - 106973118680b3 - 2026560393b2 + 7128161106*b1 - 421812793541) / 8

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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

−3.41710	−	3.41710i
−3.41710	+	3.41710i
−7.31198	+	7.31198i
−7.31198	−	7.31198i
3.21647	+	3.21647i
3.21647	−	3.21647i
−2.21647	+	2.21647i
−2.21647	−	2.21647i
8.31198	+	8.31198i
8.31198	−	8.31198i
4.41710	−	4.41710i
4.41710	+	4.41710i

−3.00000

−10.7841

−

2.94995i

−

11.9816i

9.00000

289.2

−3.00000

−10.7841

2.94995i

11.9816i

9.00000

289.3

−3.00000

−6.59835

−

9.02562i

−

28.2854i

9.00000

289.4

−3.00000

−6.59835

9.02562i

28.2854i

9.00000

289.5

−3.00000

−4.70786

−

10.1408i

21.3630i

9.00000

289.6

−3.00000

−4.70786

10.1408i

−

21.3630i

9.00000

289.7

−3.00000

4.70786

−

10.1408i

21.3630i

9.00000

289.8

−3.00000

4.70786

10.1408i

−

21.3630i

9.00000

289.9

−3.00000

6.59835

−

9.02562i

−

28.2854i

9.00000

289.10

−3.00000

6.59835

9.02562i

28.2854i

9.00000

289.11

−3.00000

10.7841

−

2.94995i

−

11.9816i

9.00000

289.12

−3.00000

10.7841

2.94995i

11.9816i

9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
20.d	odd	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	960.4.d.a	✓	12
4.b	odd	2	1		960.4.d.b	yes	12
5.b	even	2	1		960.4.d.b	yes	12
8.b	even	2	1		960.4.d.b	yes	12
8.d	odd	2	1	inner	960.4.d.a	✓	12
20.d	odd	2	1	inner	960.4.d.a	✓	12
40.e	odd	2	1		960.4.d.b	yes	12
40.f	even	2	1	inner	960.4.d.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.4.d.a	✓	12	1.a	even	1	1	trivial
960.4.d.a	✓	12	8.d	odd	2	1	inner
960.4.d.a	✓	12	20.d	odd	2	1	inner
960.4.d.a	✓	12	40.f	even	2	1	inner
960.4.d.b	yes	12	4.b	odd	2	1
960.4.d.b	yes	12	5.b	even	2	1
960.4.d.b	yes	12	8.b	even	2	1
960.4.d.b	yes	12	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(960, [\chi])$:

$ T_{7}^{6} + 1400T_{7}^{4} + 545504T_{7}^{2} + 52417600 $

$ T_{43}^{3} - 368T_{43}^{2} - 62688T_{43} + 10272960 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T + 3)^{12} $ (T + 3)^12
$5$	$ T^{12} + \cdots + 3814697265625 $ T^12 + 22T^10 + 8071T^8 - 1945900T^6 + 126109375T^4 + 5371093750*T^2 + 3814697265625
$7$	$ (T^{6} + 1400 T^{4} + \cdots + 52417600)^{2} $ (T^6 + 1400T^4 + 545504T^2 + 52417600)^2
$11$	$ (T^{6} + 4560 T^{4} + \cdots + 257281600)^{2} $ (T^6 + 4560T^4 + 2666304T^2 + 257281600)^2
$13$	$ (T^{6} - 3896 T^{4} + \cdots - 7744)^{2} $ (T^6 - 3896T^4 + 234464T^2 - 7744)^2
$17$	$ (T^{6} + 20764 T^{4} + \cdots + 236196000000)^{2} $ (T^6 + 20764T^4 + 131835600T^2 + 236196000000)^2
$19$	$ (T^{6} + 26884 T^{4} + \cdots + 234937967616)^{2} $ (T^6 + 26884T^4 + 164490624T^2 + 234937967616)^2
$23$	$ (T^{6} + 28324 T^{4} + \cdots + 475636432896)^{2} $ (T^6 + 28324T^4 + 224917504T^2 + 475636432896)^2
$29$	$ (T^{6} + \cdots + 18868250937600)^{2} $ (T^6 + 117172T^4 + 2847990096T^2 + 18868250937600)^2
$31$	$ (T^{6} - 31288 T^{4} + \cdots - 436868121600)^{2} $ (T^6 - 31288T^4 + 272990736T^2 - 436868121600)^2
$37$	$ (T^{6} - 42248 T^{4} + \cdots - 351649000000)^{2} $ (T^6 - 42248T^4 + 239439776T^2 - 351649000000)^2
$41$	$ (T^{3} - 198 T^{2} + \cdots + 4440408)^{4} $ (T^3 - 198T^2 - 55396T + 4440408)^4
$43$	$ (T^{3} - 368 T^{2} + \cdots + 10272960)^{4} $ (T^3 - 368T^2 - 62688T + 10272960)^4
$47$	$ (T^{6} + 112068 T^{4} + \cdots + 666639590400)^{2} $ (T^6 + 112068T^4 + 589078656T^2 + 666639590400)^2
$53$	$ (T^{6} + \cdots - 20194957454400)^{2} $ (T^6 - 150568T^4 + 5857066656T^2 - 20194957454400)^2
$59$	$ (T^{6} + \cdots + 22\!\cdots\!00)^{2} $ (T^6 + 598400T^4 + 82172091904T^2 + 2228402659521600)^2
$61$	$ (T^{6} + \cdots + 33\!\cdots\!16)^{2} $ (T^6 + 916384T^4 + 169798074624T^2 + 3327454931337216)^2
$67$	$ (T^{3} + 288 T^{2} + \cdots + 139828032)^{4} $ (T^3 + 288T^2 - 668736T + 139828032)^4
$71$	$ (T^{6} + \cdots - 68\!\cdots\!00)^{2} $ (T^6 - 1621728T^4 + 541138288896T^2 - 6879707136000000)^2
$73$	$ (T^{6} + \cdots + 83244456345600)^{2} $ (T^6 + 1234080T^4 + 378860177664T^2 + 83244456345600)^2
$79$	$ (T^{6} + \cdots - 26\!\cdots\!00)^{2} $ (T^6 - 2172280T^4 + 1416661303824T^2 - 268747518026649600)^2
$83$	$ (T^{3} - 304 T^{2} + \cdots + 450726336)^{4} $ (T^3 - 304T^2 - 927264T + 450726336)^4
$89$	$ (T^{3} + 778 T^{2} + \cdots - 773822440)^{4} $ (T^3 + 778T^2 - 1258468T - 773822440)^4
$97$	$ (T^{6} + \cdots + 35747006054400)^{2} $ (T^6 + 1411888T^4 + 14205102336T^2 + 35747006054400)^2
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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - 3 q^{3} - \beta_{4} q^{5} + \beta_{6} q^{7} + 9 q^{9} + ( - \beta_{11} - 2 \beta_{3}) q^{11} + ( - \beta_{10} - \beta_{5}) q^{13} + 3 \beta_{4} q^{15} + (\beta_{11} - \beta_{9} + 5 \beta_{3}) q^{17}+ \cdots + ( - 9 \beta_{11} - 18 \beta_{3}) q^{99}+O(q^{100}) \) q - 3 * q^3 - b4 * q^5 + b6 * q^7 + 9 * q^9 + (-b11 - 2b3) q^11 + (-b10 - b5) * q^13 + 3b4 q^15 + (b11 - b9 + 5b3) q^17 + (-2b11 - b9 + 2b3) * q^19 - 3b6 q^21 + (2b8 + 5b7 - 3b6 - 5b4) * q^23 + (-2b11 - b9 - 2b3 + b2 + b1 - 4) * q^25 - 27 * q^27 + (-3b8 - 11b7 + 6b6 + 11b4) * q^29 + (2b10 + 3b7 - 3b5 + 3b4) * q^31 + (3b11 + 6b3) * q^33 + (-2b9 + 14b3 - 2b2 + b1 - 24) q^35 + (-6b7 + 5b5 - 6b4) q^37 + (3b10 + 3b5) * q^39 + (-b3 - 3b2 - 2b1 + 67) * q^41 + (-2b3 - b2 - 4b1 + 123) * q^43 - 9b4 q^45 + (3b8 + 9b7 + 3b6 - 9b4) * q^47 + (2b3 + b2 + 4b1 - 124) * q^49 + (-3b11 + 3b9 - 15b3) q^51 + (-5b10 - 9b7 - 9b4) q^53 + (7b10 + 3b8 + 15b7 - 5b6 + 15b5 - 9b4) * q^55 + (6b11 + 3b9 - 6b3) q^57 + (11b11 + 2b9 - 41b3) q^59 + (-9b8 - 4b7 - 18b6 + 4b4) * q^61 + 9b6 q^63 + (-7b11 + b9 - 56b3 - 2b2 + 80) q^65 + (9b2 - 99) q^67 + (-6b8 - 15b7 + 9b6 + 15b4) * q^69 + (6b10 + 30b7 - 30b5 + 30b4) * q^71 + (-12b11 + 6b9 + 9b3) q^73 + (6b11 + 3b9 + 6b3 - 3b2 - 3b1 + 12) q^75 + (-27b10 + 2b7 - 12b5 + 2b4) * q^77 + (-20b10 + 45b7 - 9b5 + 45b4) * q^79 + 81 * q^81 + (5b3 - 2b2 + 10b1 + 102) q^83 + (-20b10 - 15b8 + 10b7 + 30b6 - 15b5 + 20b4) * q^85 + (9b8 + 33b7 - 18b6 - 33b4) * q^87 + (-7b3 - b2 - 14b1 - 259) * q^89 + (18b11 + 86b3) * q^91 + (-6b10 - 9b7 + 9b5 - 9b4) * q^93 + (-2b10 - 3b8 + 55b7 + 15b6 + 30b5 - 7b4) * q^95 + (14b11 - 2b9 + 94b3) q^97 + (-9b11 - 18b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{3} + 108 q^{9} - 44 q^{25} - 324 q^{27} - 296 q^{35} + 792 q^{41} + 1472 q^{43} - 1484 q^{49} + 952 q^{65} - 1152 q^{67} + 132 q^{75} + 972 q^{81} + 1216 q^{83} - 3112 q^{89}+O(q^{100}) \) 12 * q - 36 * q^3 + 108 * q^9 - 44 * q^25 - 324 * q^27 - 296 * q^35 + 792 * q^41 + 1472 * q^43 - 1484 * q^49 + 952 * q^65 - 1152 * q^67 + 132 * q^75 + 972 * q^81 + 1216 * q^83 - 3112 * q^89

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	960.4.d.a

Level	$960$
Weight	$4$
Character orbit	960.d
Analytic conductor	$56.642$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(56.6418336055\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - 6 x^{11} + 18 x^{10} + 300 x^{9} + 14598 x^{8} - 60928 x^{7} + 147804 x^{6} + 1450180 x^{5} + \cdots + 2737382400 \) x^12 - 6x^11 + 18x^10 + 300x^9 + 14598x^8 - 60928x^7 + 147804x^6 + 1450180x^5 + 13518601x^4 - 29643466x^3 + 68047778x^2 + 610365120*x + 2737382400
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( - 28606200764657 \nu^{11} - 107678553130919 \nu^{10} + \cdots + 85\!\cdots\!00 ) / 53\!\cdots\!00 \) (-28606200764657v^11 - 107678553130919v^10 + 14286839377994035v^9 - 324451897654674005v^8 + 639265766735073529v^7 - 183244140882525627v^6 + 124847626693067751525v^5 - 3709540294441962096995v^4 + 6559956283742582410928v^3 + 4922688670902430074346v^2 - 78070757530701486089760*v + 853114019950910061388800) / 53240300062167380256000
\(\beta_{2}\)	\(=\)	\( ( 5249469155 \nu^{11} - 145077666139 \nu^{10} + 3094091359655 \nu^{9} - 52134431764465 \nu^{8} + \cdots - 45\!\cdots\!00 ) / 30\!\cdots\!00 \) (5249469155v^11 - 145077666139v^10 + 3094091359655v^9 - 52134431764465v^8 + 232855611868085v^7 - 1618624475819007v^6 + 30133784567176305v^5 - 724640028919396135v^4 + 1367729435430712960v^3 - 787680124180438654v^2 - 13142984232319037760*v - 453190011628080643200) / 3015080986644432000
\(\beta_{3}\)	\(=\)	\( ( 6915331370263 \nu^{11} - 80794468014023 \nu^{10} + 323442252748465 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 33\!\cdots\!00 \) (6915331370263v^11 - 80794468014023v^10 + 323442252748465v^9 + 1500264010761355v^8 + 89890347406849759v^7 - 1024615208733730479v^6 + 2900643934037276355v^5 + 5359108108690675045v^4 + 46570369016402753978v^3 - 1124621592758331401498v^2 + 1358902948610911125180*v + 2074742688462266721600) / 3327518753885461266000
\(\beta_{4}\)	\(=\)	\( ( 40\!\cdots\!51 \nu^{11} + \cdots + 30\!\cdots\!00 ) / 70\!\cdots\!00 \) (40754795208499751v^11 - 485566357915119265v^10 + 5023378005722421635v^9 - 5353665270106381255v^8 + 562982360221268471213v^7 - 5135027942476227647205v^6 + 61875184181441159964765v^5 - 86826217107605079155545v^4 + 610941866986855714064236v^3 - 783402517961408900914330v^2 + 31555673942476677728776800*v + 3007772859969465191712000) / 7080959908268261574048000
\(\beta_{5}\)	\(=\)	\( ( - 11\!\cdots\!19 \nu^{11} + \cdots - 78\!\cdots\!00 ) / 64\!\cdots\!00 \) (-1179207355312919v^11 + 5055403466864107v^10 + 2250309511779445v^9 - 435564475626491735v^8 - 17724183040865622857v^7 + 45608956902906922431v^6 + 135625619458107816675v^5 - 2403707765863503043265v^4 - 16980502844995568071624v^3 + 21371140519339045256062v^2 + 238637388409374724956480*v - 783064564813946719526400) / 64962934938240931872000
\(\beta_{6}\)	\(=\)	\( ( 19\!\cdots\!93 \nu^{11} + \cdots + 72\!\cdots\!00 ) / 87\!\cdots\!00 \) (1934372922803593v^11 - 12716860284760433v^10 + 59114914395710965v^9 + 473792237723756605v^8 + 28165174889944007599v^7 - 129253959831677193309v^6 + 635931766652094328755v^5 + 1833658999221919156795v^4 + 25037631342969820921808v^3 - 47264922583194723659258v^2 + 529212600963997979429280*v + 724291611935477269593600) / 87419258126768661408000
\(\beta_{7}\)	\(=\)	\( ( - 16\!\cdots\!19 \nu^{11} + \cdots - 66\!\cdots\!00 ) / 70\!\cdots\!00 \) (-16556965419423019v^11 + 92306962583284286v^10 - 254100039576323155v^9 - 5318511198810265000v^8 - 236425806837572067127v^7 + 882342486613512256698v^6 - 2016969655917771927105v^5 - 26448337452075259946500v^4 - 131917787806858289245154v^3 + 248686201635138574053116v^2 - 1629999844090684794607440*v - 6644713394874573633225600) / 708095990826826157404800
\(\beta_{8}\)	\(=\)	\( ( 215567487415101 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 48\!\cdots\!00 \) (215567487415101v^11 - 1523965074430661v^10 + 9066039751198105v^9 + 47280778653465985v^8 + 3003716617850958043v^7 - 14910159110254833953v^6 + 101495732455616332335v^5 + 167403641542706185815v^4 + 1237428709276880787856v^3 - 1547989058908558208386v^2 + 41103419556176793782560*v + 64617229350542493427200) / 4856625451487147856000
\(\beta_{9}\)	\(=\)	\( ( - 12\!\cdots\!61 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 26\!\cdots\!00 \) (-1208724988289461v^11 + 18551517427273769v^10 - 78057188646759865v^9 - 222065911156639405v^8 - 13890064213629217363v^7 + 232139909686647025557v^6 - 668068950346366078095v^5 - 681606620985336775195v^4 + 1567425413914970049424v^3 + 78970772770666714537274v^2 - 120421318072287770131440*v - 252396365944789055980800) / 26620150031083690128000
\(\beta_{10}\)	\(=\)	\( ( - 169860038210899 \nu^{11} + 661777164565559 \nu^{10} + \cdots - 99\!\cdots\!00 ) / 36\!\cdots\!00 \) (-169860038210899v^11 + 661777164565559v^10 + 1907527771683305v^9 - 71667946412210515v^8 - 2490482561363103757v^7 + 5797301894803039707v^6 + 37448808346181987535v^5 - 408998372404918971685v^4 - 1719387721142576106344v^3 + 2699941523000578935734v^2 + 27659641534183733444160*v - 99635781138591912652800) / 3609051941013385104000
\(\beta_{11}\)	\(=\)	\( ( 26\!\cdots\!45 \nu^{11} + \cdots + 64\!\cdots\!00 ) / 53\!\cdots\!00 \) (2628210621539645v^11 - 36176790982136089v^10 + 149504680672139705v^9 + 519887465865230285v^8 + 31971951826365576755v^7 - 458238056093189879157v^6 + 1310738298755515797855v^5 + 1715822323522946653115v^4 + 5372109912204030857800v^3 - 278590875798475619024554v^2 + 368426352800655831596640*v + 649718073381682590220800) / 53240300062167380256000

\(\nu\)	\(=\)	\( ( \beta_{10} - \beta_{8} - 4\beta_{7} - \beta_{3} + 4 ) / 8 \) (b10 - b8 - 4*b7 - b3 + 4) / 8
\(\nu^{2}\)	\(=\)	\( ( 4\beta_{11} + 2\beta_{9} - \beta_{8} - 2\beta_{7} + 2\beta_{4} - 58\beta_{3} ) / 4 \) (4b11 + 2b9 - b8 - 2b7 + 2b4 - 58*b3) / 4
\(\nu^{3}\)	\(=\)	\( ( 12 \beta_{11} - 90 \beta_{10} + 6 \beta_{9} - 73 \beta_{8} - 34 \beta_{7} - 90 \beta_{6} + 90 \beta_{5} + \cdots - 671 ) / 8 \) (12b11 - 90b10 + 6b9 - 73b8 - 34b7 - 90b6 + 90b5 + 326b4 - 170b3 - 3b2 + 6*b1 - 671) / 8
\(\nu^{4}\)	\(=\)	\( ( -178\beta_{10} + 288\beta_{7} + 180\beta_{5} + 288\beta_{4} + 159\beta_{3} - 111\beta_{2} + 318\beta _1 - 22043 ) / 4 \) (-178b10 + 288b7 + 180b5 + 288b4 + 159b3 - 111b2 + 318*b1 - 22043) / 4
\(\nu^{5}\)	\(=\)	\( ( - 2660 \beta_{11} - 10764 \beta_{10} - 1570 \beta_{9} + 7499 \beta_{8} + 36526 \beta_{7} + \cdots - 107981 ) / 8 \) (-2660b11 - 10764b10 - 1570b9 + 7499b8 + 36526b7 + 13530b6 + 13530b5 - 6530b4 + 28974b3 - 545b2 + 1570*b1 - 107981) / 8
\(\nu^{6}\)	\(=\)	\( ( - 68416 \beta_{11} - 42608 \beta_{9} + 21779 \beta_{8} + 62808 \beta_{7} + 39690 \beta_{6} + \cdots + 687052 \beta_{3} ) / 4 \) (-68416b11 - 42608b9 + 21779b8 + 62808b7 + 39690b6 - 62808b4 + 687052*b3) / 4
\(\nu^{7}\)	\(=\)	\( ( - 460348 \beta_{11} + 1362080 \beta_{10} - 287294 \beta_{9} + 897887 \beta_{8} + 928386 \beta_{7} + \cdots + 17617219 ) / 8 \) (-460348b11 + 1362080b10 - 287294b9 + 897887b8 + 928386b7 + 1789050b6 - 1789050b5 - 4519934b4 + 4469200b3 + 86527b2 - 287294*b1 + 17617219) / 8
\(\nu^{8}\)	\(=\)	\( ( 5156872 \beta_{10} - 6779232 \beta_{7} - 6787440 \beta_{5} - 6779232 \beta_{4} - 2775691 \beta_{3} + \cdots + 331530127 ) / 4 \) (5156872b10 - 6779232b7 - 6787440b5 - 6779232b4 - 2775691b3 + 1603579b2 - 5551382*b1 + 331530127) / 4
\(\nu^{9}\)	\(=\)	\( ( 73347180 \beta_{11} + 175719976 \beta_{10} + 46541190 \beta_{9} - 113872741 \beta_{8} + \cdots + 2774167879 ) / 8 \) (73347180b11 + 175719976b10 + 46541190b9 - 113872741b8 - 579185434b7 - 233157390b6 - 233157390b5 + 123694470b4 - 750135406b3 + 13402995b2 - 46541190*b1 + 2774167879) / 8
\(\nu^{10}\)	\(=\)	\( ( 1136905264 \beta_{11} + 724446632 \beta_{9} - 519123541 \beta_{8} - 1604238032 \beta_{7} + \cdots - 11242050388 \beta_{3} ) / 4 \) (1136905264b11 + 724446632b9 - 519123541b8 - 1604238032b7 - 1065079470b6 + 1604238032b4 - 11242050388*b3) / 4
\(\nu^{11}\)	\(=\)	\( ( 11181281892 \beta_{11} - 22961104200 \beta_{10} + 7128161106 \beta_{9} - 14804378353 \beta_{8} + \cdots - 421812793541 ) / 8 \) (11181281892b11 - 22961104200b10 + 7128161106b9 - 14804378353b8 - 16313451694b7 - 30545297910b6 + 30545297910b5 + 75530965106b4 - 106973118680b3 - 2026560393b2 + 7128161106*b1 - 421812793541) / 8

\(n\)	\(511\)	\(577\)	\(641\)	\(901\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 289.12
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T + 3)^{12} \) (T + 3)^12
$5$	\( T^{12} + \cdots + 3814697265625 \) T^12 + 22T^10 + 8071T^8 - 1945900T^6 + 126109375T^4 + 5371093750*T^2 + 3814697265625
$7$	\( (T^{6} + 1400 T^{4} + \cdots + 52417600)^{2} \) (T^6 + 1400T^4 + 545504T^2 + 52417600)^2
$11$	\( (T^{6} + 4560 T^{4} + \cdots + 257281600)^{2} \) (T^6 + 4560T^4 + 2666304T^2 + 257281600)^2
$13$	\( (T^{6} - 3896 T^{4} + \cdots - 7744)^{2} \) (T^6 - 3896T^4 + 234464T^2 - 7744)^2
$17$	\( (T^{6} + 20764 T^{4} + \cdots + 236196000000)^{2} \) (T^6 + 20764T^4 + 131835600T^2 + 236196000000)^2
$19$	\( (T^{6} + 26884 T^{4} + \cdots + 234937967616)^{2} \) (T^6 + 26884T^4 + 164490624T^2 + 234937967616)^2
$23$	\( (T^{6} + 28324 T^{4} + \cdots + 475636432896)^{2} \) (T^6 + 28324T^4 + 224917504T^2 + 475636432896)^2
$29$	\( (T^{6} + \cdots + 18868250937600)^{2} \) (T^6 + 117172T^4 + 2847990096T^2 + 18868250937600)^2
$31$	\( (T^{6} - 31288 T^{4} + \cdots - 436868121600)^{2} \) (T^6 - 31288T^4 + 272990736T^2 - 436868121600)^2
$37$	\( (T^{6} - 42248 T^{4} + \cdots - 351649000000)^{2} \) (T^6 - 42248T^4 + 239439776T^2 - 351649000000)^2
$41$	\( (T^{3} - 198 T^{2} + \cdots + 4440408)^{4} \) (T^3 - 198T^2 - 55396T + 4440408)^4
$43$	\( (T^{3} - 368 T^{2} + \cdots + 10272960)^{4} \) (T^3 - 368T^2 - 62688T + 10272960)^4
$47$	\( (T^{6} + 112068 T^{4} + \cdots + 666639590400)^{2} \) (T^6 + 112068T^4 + 589078656T^2 + 666639590400)^2
$53$	\( (T^{6} + \cdots - 20194957454400)^{2} \) (T^6 - 150568T^4 + 5857066656T^2 - 20194957454400)^2
$59$	\( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) (T^6 + 598400T^4 + 82172091904T^2 + 2228402659521600)^2
$61$	\( (T^{6} + \cdots + 33\!\cdots\!16)^{2} \) (T^6 + 916384T^4 + 169798074624T^2 + 3327454931337216)^2
$67$	\( (T^{3} + 288 T^{2} + \cdots + 139828032)^{4} \) (T^3 + 288T^2 - 668736T + 139828032)^4
$71$	\( (T^{6} + \cdots - 68\!\cdots\!00)^{2} \) (T^6 - 1621728T^4 + 541138288896T^2 - 6879707136000000)^2
$73$	\( (T^{6} + \cdots + 83244456345600)^{2} \) (T^6 + 1234080T^4 + 378860177664T^2 + 83244456345600)^2
$79$	\( (T^{6} + \cdots - 26\!\cdots\!00)^{2} \) (T^6 - 2172280T^4 + 1416661303824T^2 - 268747518026649600)^2
$83$	\( (T^{3} - 304 T^{2} + \cdots + 450726336)^{4} \) (T^3 - 304T^2 - 927264T + 450726336)^4
$89$	\( (T^{3} + 778 T^{2} + \cdots - 773822440)^{4} \) (T^3 + 778T^2 - 1258468T - 773822440)^4
$97$	\( (T^{6} + \cdots + 35747006054400)^{2} \) (T^6 + 1411888T^4 + 14205102336T^2 + 35747006054400)^2
show more
show less