LMFDB - Newform orbit 10000.2.a.bq

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$10000 = 2^{4} \cdot 5^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

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:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80$ x^16 - 4x^15 - 26x^14 + 110x^13 + 250x^12 - 1154x^11 - 1074x^10 + 5784x^9 + 1890x^8 - 14210x^7 - 726x^6 + 15974x^5 + 61x^4 - 7820x^3 - 180x^2 + 1360*x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$5^{3}$
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\ldots,\beta_{15}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + (\beta_{8} - 1) q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{5} + 1) q^{11} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 1) q^{13} + ( - \beta_{12} + \beta_{8} + \cdots - \beta_1) q^{17}+ \cdots + ( - \beta_{14} + 2 \beta_{13} + \cdots + 3) q^{99}+O(q^{100})$ q - b1 * q^3 + (b8 - 1) * q^7 + (b2 + 1) * q^9 + (b5 + 1) * q^11 + (-b11 + b10 + b9 - b5 - b3 + 1) * q^13 + (-b12 + b8 + b7 - b5 - b4 - b1) * q^17 + (b12 + b11 - b10 - b9 + b5 + b4 + b2 + 1) * q^19 + (-b15 - b13 - b8 + b7 - b4 - b2 + b1 + 1) * q^21 + (b15 + b12 - b11 + b10 + b9 - b8 + b4 - 2b3 + b1 + 1) q^23 + (-b14 + b13 - b12 + b8 + b7 - b6 - b4 - b2 - b1 - 2) * q^27 + (-b12 - b9 + b6 - b4 + 1) * q^29 + (-b15 - b13 - b10 - b9 - 2b4 - 2b1) * q^31 + (-b14 + b12 - b9 - b6 + b2 - 2b1 + 1) q^33 + (b15 + b10 + b9 + b8 + 1) * q^37 + (-b14 - b12 + b10 + 2b8 + b7 + b6 + 2b3 - b2 - b1 - 1) * q^39 + (b13 + b12 - b10 - b6 + b5 + 2b4 + b2 + 2b1 + 1) * q^41 + (b14 - b9 + b6 + b4 + b2 - 1) * q^43 + (-b15 + b14 - b13 - b12 + b10 + b9 - b6 - b5 - 2b4 - b2 - b1 - 2) q^47 + (b15 + b12 + b9 - 2b8 + b5 - 2b3 + 2b1 + 4) q^49 + (b14 - b12 + b9 + b8 - b6 - b5 - b4 + b2 - 2b1 + 1) q^51 + (b14 + b12 - b10 - b7 - b6 + 2b4 - b3 + b2 + b1 + 1) q^53 + (-b15 + b13 - b12 - b10 + 2b6 + b5 - b1 + 3) q^57 + (b15 + 2b12 - b11 + b9 - b8 - b7 - b5 + 2b4 - 2b3 + 2b2 - b1 + 4) * q^59 + (b10 + b9 - b8 - b7 - b6 + b4 + b1 + 1) * q^61 + (-b14 - b11 + b9 + 2b8 - b7 + 3b6 - b5 + b4 + 2b3 - b2 - b1 - 4) q^63 + (-b15 - b14 - 2b13 - b10 + b7 - b6 - b5 - 3b4 - b2 - 2b1 - 4) q^67 + (b15 - b14 + b13 - b11 + b10 + 2b8 - b7 + b6 + b4 + 2b3 + b2 + b1 - 1) * q^69 + (-b15 + b14 + b13 - b12 + b11 + 2b10 - b8 - b6 + b5 - b4 - b2 + 1) q^71 + (b14 + b11 - b10 - b6 + b5 + b4 + b3 - b2 + 2) * q^73 + (-b13 + b12 - b9 + b8 - b7 + 4b6 - b4 - b1 + 3) q^77 + (b15 + 2b14 - b12 - b11 - b8 - b7 + 2b6 + 3b4 - 2b3 + b2 + b1 + 3) * q^79 + (b15 + b14 + b12 + b11 + b10 - b9 - b8 - 2b7 - b6 + b5 + 2b4 + 2b2 + 2b1 + 2) * q^81 + (-b15 - b12 + b11 - b10 - b9 + 2b8 + 4b6 + 2b3 - b1 - 2) q^83 + (b15 + b14 + 2b13 - b12 + b10 - b7 - 3b6 + b5 + 2b4 + b2 - 4) q^87 + (-b15 - 2b13 + 3b11 - 2b10 - b9 + b8 + b7 + b6 + b5 - 2b4 + 2b3 - b2 - b1 + 1) q^89 + (b15 - 2b14 + 2b13 + b11 + b9 + 3b8 + b7 - 2b6 + 2b3 - b1 - 1) q^91 + (-b12 + b11 + 2b8 + b7 - 4b6 + b5 + b4 + 2b3 - 1) q^93 + (-b12 - b7 + 2b6 - b5 - b4 - 3b1 + 3) * q^97 + (-b14 + 2b13 - 2b12 - 2b11 + 2b10 + 2b9 + b8 + b6 - b5 - 2b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 4 q^{3} - 8 q^{7} + 20 q^{9} + 12 q^{11} + 10 q^{13} + 8 q^{17} + 12 q^{19} + 8 q^{21} - 12 q^{23} - 22 q^{27} + 16 q^{29} + 2 q^{31} + 24 q^{33} + 22 q^{37} + 4 q^{39} + 20 q^{41} - 26 q^{43} - 24 q^{47}+ \cdots + 46 q^{99}+O(q^{100})$ 16 * q - 4 * q^3 - 8 * q^7 + 20 * q^9 + 12 * q^11 + 10 * q^13 + 8 * q^17 + 12 * q^19 + 8 * q^21 - 12 * q^23 - 22 * q^27 + 16 * q^29 + 2 * q^31 + 24 * q^33 + 22 * q^37 + 4 * q^39 + 20 * q^41 - 26 * q^43 - 24 * q^47 + 30 * q^49 + 30 * q^51 + 16 * q^53 + 24 * q^57 + 40 * q^59 + 22 * q^61 - 52 * q^63 - 50 * q^67 + 22 * q^69 + 4 * q^71 + 32 * q^73 + 30 * q^77 + 8 * q^79 + 44 * q^81 - 30 * q^83 - 48 * q^87 + 28 * q^89 + 32 * q^91 + 40 * q^93 + 36 * q^97 + 46 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80$ x^16 - 4*x^15 - 26*x^14 + 110*x^13 + 250*x^12 - 1154*x^11 - 1074*x^10 + 5784*x^9 + 1890*x^8 - 14210*x^7 - 726*x^6 + 15974*x^5 + 61*x^4 - 7820*x^3 - 180*x^2 + 1360*x + 80 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 4$ v^2 - 4
$\beta_{3}$	$=$	$( 177 \nu^{15} - 6961 \nu^{14} + 34464 \nu^{13} + 147458 \nu^{12} - 1048624 \nu^{11} - 886366 \nu^{10} + \cdots + 2503340 ) / 624200$ (177v^15 - 6961v^14 + 34464v^13 + 147458v^12 - 1048624v^11 - 886366v^10 + 11507128v^9 - 402388v^8 - 59298466v^7 + 18139998v^6 + 144978344v^5 - 42106782v^4 - 143470745v^3 + 9643645v^2 + 38618390*v + 2503340) / 624200
$\beta_{4}$	$=$	$( - 3320 \nu^{15} + 7068 \nu^{14} + 106035 \nu^{13} - 192200 \nu^{12} - 1354885 \nu^{11} + \cdots + 651760 ) / 312100$ (-3320v^15 + 7068v^14 + 106035v^13 - 192200v^12 - 1354885v^11 + 1971910v^10 + 8807743v^9 - 9491280v^8 - 30557465v^7 + 21663380v^6 + 54404455v^5 - 20568758v^4 - 44267305v^3 + 4690680v^2 + 11567080*v + 651760) / 312100
$\beta_{5}$	$=$	$( 16083 \nu^{15} - 40628 \nu^{14} - 491454 \nu^{13} + 1092422 \nu^{12} + 6003374 \nu^{11} + \cdots - 5172720 ) / 624200$ (16083v^15 - 40628v^14 - 491454v^13 + 1092422v^12 + 6003374v^11 - 11027574v^10 - 37521682v^9 + 51608148v^8 + 127372626v^7 - 110925978v^6 - 230747714v^5 + 90053546v^4 + 204749835v^3 - 11172960v^2 - 58292880*v - 5172720) / 624200
$\beta_{6}$	$=$	$( 8147 \nu^{15} - 29268 \nu^{14} - 218890 \nu^{13} + 790135 \nu^{12} + 2228950 \nu^{11} + \cdots - 799260 ) / 312100$ (8147v^15 - 29268v^14 - 218890v^13 + 790135v^12 + 2228950v^11 - 8046753v^10 - 10721788v^9 + 38314505v^8 + 24889110v^7 - 85211405v^6 - 27578102v^5 + 75735723v^4 + 21065725v^3 - 19442235v^2 - 6157140*v - 799260) / 312100
$\beta_{7}$	$=$	$( - 34985 \nu^{15} + 160082 \nu^{14} + 828954 \nu^{13} - 4334162 \nu^{12} - 6655254 \nu^{11} + \cdots - 11782400 ) / 1248400$ (-34985v^15 + 160082v^14 + 828954v^13 - 4334162v^12 - 6655254v^11 + 44355342v^10 + 17636926v^9 - 212702468v^8 + 18321894v^7 + 477154238v^6 - 122193254v^5 - 431363050v^4 + 64482055v^3 + 138862130v^2 - 2942820*v - 11782400) / 1248400
$\beta_{8}$	$=$	$( - 9952 \nu^{15} + 47268 \nu^{14} + 222753 \nu^{13} - 1261704 \nu^{12} - 1520633 \nu^{11} + \cdots - 1490220 ) / 312100$ (-9952v^15 + 47268v^14 + 222753v^13 - 1261704v^12 - 1520633v^11 + 12633462v^10 + 899021v^9 - 58477396v^8 + 27279413v^7 + 123277476v^6 - 92585801v^5 - 98707854v^4 + 82051325v^3 + 29185780v^2 - 19836740*v - 1490220) / 312100
$\beta_{9}$	$=$	$( 19853 \nu^{15} - 110886 \nu^{14} - 371328 \nu^{13} + 2921034 \nu^{12} + 1061908 \nu^{11} + \cdots + 5150560 ) / 624200$ (19853v^15 - 110886v^14 - 371328v^13 + 2921034v^12 + 1061908v^11 - 28644586v^10 + 18264116v^9 + 127754076v^8 - 149557988v^7 - 248881846v^6 + 393947300v^5 + 160654902v^4 - 345096785v^3 - 42669950v^2 + 89851580*v + 5150560) / 624200
$\beta_{10}$	$=$	$( - 45159 \nu^{15} + 195662 \nu^{14} + 1091690 \nu^{13} - 5247810 \nu^{12} - 9150790 \nu^{11} + \cdots + 1927440 ) / 1248400$ (-45159v^15 + 195662v^14 + 1091690v^13 - 5247810v^12 - 9150790v^11 + 52916166v^10 + 28050182v^9 - 247591200v^8 + 1748630v^7 + 530880510v^6 - 120398966v^5 - 432532962v^4 + 64304365v^3 + 105387530v^2 - 80660*v + 1927440) / 1248400
$\beta_{11}$	$=$	$( 48227 \nu^{15} - 237254 \nu^{14} - 1043610 \nu^{13} + 6309830 \nu^{12} + 6421510 \nu^{11} + \cdots + 13166720 ) / 1248400$ (48227v^15 - 237254v^14 - 1043610v^13 + 6309830v^12 + 6421510v^11 - 62832658v^10 + 5003306v^9 + 288134740v^8 - 174817550v^7 - 596129770v^6 + 537759398v^5 + 455404854v^4 - 473959465v^3 - 131899470v^2 + 125829500*v + 13166720) / 1248400
$\beta_{12}$	$=$	$( - 17507 \nu^{15} + 57768 \nu^{14} + 498354 \nu^{13} - 1586982 \nu^{12} - 5557734 \nu^{11} + \cdots + 1137920 ) / 312100$ (-17507v^15 + 57768v^14 + 498354v^13 - 1586982v^12 - 5557734v^11 + 16581630v^10 + 31055632v^9 - 82178728v^8 - 93025186v^7 + 195524538v^6 + 151506798v^5 - 196444706v^4 - 132328025v^3 + 57669720v^2 + 37521800*v + 1137920) / 312100
$\beta_{13}$	$=$	$( - 70841 \nu^{15} + 216442 \nu^{14} + 2080246 \nu^{13} - 5960498 \nu^{12} - 24203026 \nu^{11} + \cdots + 10337280 ) / 1248400$ (-70841v^15 + 216442v^14 + 2080246v^13 - 5960498v^12 - 24203026v^11 + 62443582v^10 + 143082938v^9 - 310339532v^8 - 459528014v^7 + 740206462v^6 + 800601470v^5 - 741410474v^4 - 710243405v^3 + 201556050v^2 + 202428540*v + 10337280) / 1248400
$\beta_{14}$	$=$	$( - 47457 \nu^{15} + 211662 \nu^{14} + 1129108 \nu^{13} - 5692644 \nu^{12} - 9103068 \nu^{11} + \cdots - 4435440 ) / 624200$ (-47457v^15 + 211662v^14 + 1129108v^13 - 5692644v^12 - 9103068v^11 + 57652812v^10 + 23874800v^9 - 271764786v^8 + 31342848v^7 + 591282276v^6 - 202633796v^5 - 501246988v^4 + 152905385v^3 + 142120120v^2 - 30163500*v - 4435440) / 624200
$\beta_{15}$	$=$	$( 29696 \nu^{15} - 104425 \nu^{14} - 808595 \nu^{13} + 2827855 \nu^{12} + 8411315 \nu^{11} + \cdots - 2334800 ) / 312100$ (29696v^15 - 104425v^14 - 808595v^13 + 2827855v^12 + 8411315v^11 - 28916859v^10 - 41983295v^9 + 138466635v^8 + 104600085v^7 - 310514865v^6 - 132251941v^5 + 279446865v^4 + 107258485v^3 - 71490140v^2 - 31028680*v - 2334800) / 312100

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 4$ b2 + 4
$\nu^{3}$	$=$	$\beta_{14} - \beta_{13} + \beta_{12} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + 7\beta _1 + 2$ b14 - b13 + b12 - b8 - b7 + b6 + b4 + b2 + 7*b1 + 2
$\nu^{4}$	$=$	$\beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots + 29$ b15 + b14 + b12 + b11 + b10 - b9 - b8 - 2b7 - b6 + b5 + 2b4 + 11b2 + 2b1 + 29
$\nu^{5}$	$=$	$10 \beta_{14} - 13 \beta_{13} + 12 \beta_{12} + 3 \beta_{11} - \beta_{9} - 7 \beta_{8} - 10 \beta_{7} + \cdots + 24$ 10b14 - 13b13 + 12b12 + 3b11 - b9 - 7b8 - 10b7 + 10b6 - b5 + 12b4 + 2b3 + 14b2 + 57*b1 + 24
$\nu^{6}$	$=$	$18 \beta_{15} + 15 \beta_{14} - \beta_{13} + 17 \beta_{12} + 18 \beta_{11} + 12 \beta_{10} - 19 \beta_{9} + \cdots + 239$ 18b15 + 15b14 - b13 + 17b12 + 18b11 + 12b10 - 19b9 - 12b8 - 29b7 - 19b6 + 14b5 + 37b4 + 4b3 + 115b2 + 37b1 + 239
$\nu^{7}$	$=$	$8 \beta_{15} + 90 \beta_{14} - 140 \beta_{13} + 128 \beta_{12} + 54 \beta_{11} + 2 \beta_{10} + \cdots + 258$ 8b15 + 90b14 - 140b13 + 128b12 + 54b11 + 2b10 - 28b9 - 42b8 - 96b7 + 76b6 - 10b5 + 148b4 + 44b3 + 172b2 + 507*b1 + 258
$\nu^{8}$	$=$	$238 \beta_{15} + 178 \beta_{14} - 36 \beta_{13} + 214 \beta_{12} + 256 \beta_{11} + 114 \beta_{10} + \cdots + 2068$ 238b15 + 178b14 - 36b13 + 214b12 + 256b11 + 114b10 - 264b9 - 104b8 - 330b7 - 294b6 + 158b5 + 518b4 + 104b3 + 1189b2 + 514*b1 + 2068
$\nu^{9}$	$=$	$212 \beta_{15} + 817 \beta_{14} - 1431 \beta_{13} + 1323 \beta_{12} + 754 \beta_{11} + 56 \beta_{10} + \cdots + 2674$ 212b15 + 817b14 - 1431b13 + 1323b12 + 754b11 + 56b10 - 482b9 - 203b8 - 949b7 + 413b6 - 38b5 + 1869b4 + 684b3 + 2037b2 + 4799*b1 + 2674
$\nu^{10}$	$=$	$2885 \beta_{15} + 1975 \beta_{14} - 708 \beta_{13} + 2471 \beta_{12} + 3295 \beta_{11} + 1033 \beta_{10} + \cdots + 18415$ 2885b15 + 1975b14 - 708b13 + 2471b12 + 3295b11 + 1033b10 - 3277b9 - 721b8 - 3520b7 - 4095b6 + 1713b5 + 6674b4 + 1792b3 + 12333b2 + 6504*b1 + 18415
$\nu^{11}$	$=$	$3776 \beta_{15} + 7650 \beta_{14} - 14421 \beta_{13} + 13548 \beta_{12} + 9679 \beta_{11} + 960 \beta_{10} + \cdots + 27222$ 3776b15 + 7650b14 - 14421b13 + 13548b12 + 9679b11 + 960b10 - 6933b9 - 193b8 - 9678b7 + 2b6 + 563b5 + 23434b4 + 9354b3 + 23750b2 + 47439*b1 + 27222
$\nu^{12}$	$=$	$33886 \beta_{15} + 21335 \beta_{14} - 10963 \beta_{13} + 27629 \beta_{12} + 40274 \beta_{11} + \cdots + 167249$ 33886b15 + 21335b14 - 10963b13 + 27629b12 + 40274b11 + 9436b10 - 38751b9 - 3062b8 - 36947b7 - 53403b6 + 18590b5 + 82901b4 + 26260b3 + 129039b2 + 78939*b1 + 167249
$\nu^{13}$	$=$	$56916 \beta_{15} + 74056 \beta_{14} - 145338 \beta_{13} + 138764 \beta_{12} + 119414 \beta_{11} + \cdots + 274612$ 56916b15 + 74056b14 - 145338b13 + 138764b12 + 119414b11 + 13422b10 - 91388b9 + 15426b8 - 101250b7 - 46238b6 + 16746b5 + 288936b4 + 120292b3 + 274638b2 + 483503*b1 + 274612
$\nu^{14}$	$=$	$392784 \beta_{15} + 227784 \beta_{14} - 150324 \beta_{13} + 304908 \beta_{12} + 478292 \beta_{11} + \cdots + 1541960$ 392784b15 + 227784b14 - 150324b13 + 304908b12 + 478292b11 + 88660b10 - 448124b9 + 18652b8 - 388348b7 - 667204b6 + 203964b5 + 1008308b4 + 354416b3 + 1364309b2 + 936868*b1 + 1541960
$\nu^{15}$	$=$	$784500 \beta_{15} + 738565 \beta_{14} - 1473625 \beta_{13} + 1428221 \beta_{12} + 1439528 \beta_{11} + \cdots + 2758914$ 784500b15 + 738565b14 - 1473625b13 + 1428221b12 + 1439528b11 + 169292b10 - 1146904b9 + 332039b8 - 1080725b7 - 969235b6 + 288836b5 + 3503613b4 + 1494400b3 + 3160061b2 + 5038923*b1 + 2758914

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

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}

1.1

3.36926
3.07129
2.93924
2.30769
1.62428
1.56513
0.720900
0.686530
−0.0595607
−0.631998
−0.764364
−0.853232
−2.02783
−2.24082
−2.71029
−2.99623

−3.36926

−0.794375

8.35194

1.2

−3.07129

−4.49756

6.43283

1.3

−2.93924

3.13612

5.63915

1.4

−2.30769

−4.74404

2.32542

1.5

−1.62428

−2.51754

−0.361706

1.6

−1.56513

−0.0338937

−0.550359

1.7

−0.720900

4.42421

−2.48030

1.8

−0.686530

3.54998

−2.52868

1.9

0.0595607

−1.71675

−2.99645

1.10

0.631998

2.09441

−2.60058

1.11

0.764364

−4.32704

−2.41575

1.12

0.853232

−1.47738

−2.27199

1.13

2.02783

0.498275

1.11208

1.14

2.24082

1.85909

2.02129

1.15

2.71029

0.760910

4.34569

1.16

2.99623

−4.21442

5.97742

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	10000.2.a.bq		16
4.b	odd	2	1		5000.2.a.r		16
5.b	even	2	1		10000.2.a.br		16
20.d	odd	2	1		5000.2.a.q		16
25.f	odd	20	2		400.2.y.d		32
100.h	odd	10	2		1000.2.m.e		32
100.j	odd	10	2		1000.2.m.d		32
100.l	even	20	2		200.2.q.a	✓	32
100.l	even	20	2		1000.2.q.c		32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.q.a	✓	32	100.l	even	20	2
400.2.y.d		32	25.f	odd	20	2
1000.2.m.d		32	100.j	odd	10	2
1000.2.m.e		32	100.h	odd	10	2
1000.2.q.c		32	100.l	even	20	2
5000.2.a.q		16	20.d	odd	2	1
5000.2.a.r		16	4.b	odd	2	1
10000.2.a.bq		16	1.a	even	1	1	trivial
10000.2.a.br		16	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(10000))$ :

T_{3}^{16} + 4 T_{3}^{15} - 26 T_{3}^{14} - 110 T_{3}^{13} + 250 T_{3}^{12} + 1154 T_{3}^{11} + \cdots + 80

T_{7}^{16} + 8 T_{7}^{15} - 39 T_{7}^{14} - 416 T_{7}^{13} + 376 T_{7}^{12} + 8084 T_{7}^{11} + \cdots - 4864

T_{11}^{16} - 12 T_{11}^{15} - 31 T_{11}^{14} + 794 T_{11}^{13} - 756 T_{11}^{12} - 18090 T_{11}^{11} + \cdots - 112384

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16}$ T^16
$3$	$T^{16} + 4 T^{15} + \cdots + 80$ T^16 + 4T^15 - 26T^14 - 110T^13 + 250T^12 + 1154T^11 - 1074T^10 - 5784T^9 + 1890T^8 + 14210T^7 - 726T^6 - 15974T^5 + 61T^4 + 7820T^3 - 180T^2 - 1360*T + 80
$5$	$T^{16}$ T^16
$7$	$T^{16} + 8 T^{15} + \cdots - 4864$ T^16 + 8T^15 - 39T^14 - 416T^13 + 376T^12 + 8084T^11 + 2001T^10 - 73936T^9 - 49339T^8 + 328084T^7 + 254484T^6 - 666512T^5 - 471616T^4 + 518336T^3 + 211264T^2 - 136960*T - 4864
$11$	$T^{16} - 12 T^{15} + \cdots - 112384$ T^16 - 12T^15 - 31T^14 + 794T^13 - 756T^12 - 18090T^11 + 30797T^10 + 201148T^9 - 313887T^8 - 1295072T^7 + 1146112T^6 + 4721600T^5 - 110496T^4 - 7072256T^3 - 5263616T^2 - 1359872*T - 112384
$13$	$T^{16} - 10 T^{15} + \cdots - 9030899$ T^16 - 10T^15 - 69T^14 + 948T^13 + 789T^12 - 32134T^11 + 31266T^10 + 498422T^9 - 839634T^8 - 3840022T^7 + 7398634T^6 + 14322102T^5 - 25984359T^4 - 22084732T^3 + 27705339T^2 + 10386994*T - 9030899
$17$	$T^{16} - 8 T^{15} + \cdots + 6010000$ T^16 - 8T^15 - 131T^14 + 1188T^13 + 5586T^12 - 63960T^11 - 77235T^10 + 1589020T^9 - 418830T^8 - 18886200T^7 + 16005225T^6 + 100396100T^5 - 78428475T^4 - 197563000T^3 - 15218500T^2 + 48960000*T + 6010000
$19$	$T^{16} - 12 T^{15} + \cdots + 2808976$ T^16 - 12T^15 - 101T^14 + 1694T^13 + 1594T^12 - 89720T^11 + 152467T^10 + 2147458T^9 - 7288022T^8 - 19883152T^7 + 116811047T^6 - 29456570T^5 - 616290291T^4 + 1059821384T^3 - 316350596T^2 - 284973632*T + 2808976
$23$	$T^{16} + \cdots + 6486139136$ T^16 + 12T^15 - 149T^14 - 2004T^13 + 8651T^12 + 131036T^11 - 275914T^10 - 4335444T^9 + 6130351T^8 + 77947936T^7 - 98619081T^6 - 745187688T^5 + 937157309T^4 + 3460786664T^3 - 4176098976T^2 - 6153971840*T + 6486139136
$29$	$T^{16} - 16 T^{15} + \cdots + 35290081$ T^16 - 16T^15 - 69T^14 + 2178T^13 - 3051T^12 - 103590T^11 + 370318T^10 + 1919386T^9 - 10566762T^8 - 6634746T^7 + 103070078T^6 - 114781450T^5 - 175590171T^4 + 325618662T^3 - 39346429T^2 - 117442904*T + 35290081
$31$	$T^{16} + \cdots + 2080762000$ T^16 - 2T^15 - 241T^14 + 562T^13 + 21831T^12 - 66040T^11 - 921330T^10 + 3755160T^9 + 16598575T^8 - 101567550T^7 - 21148125T^6 + 1049512550T^5 - 2163357375T^4 + 109261000T^3 + 4584827000T^2 - 5546968000*T + 2080762000
$37$	$T^{16} + \cdots + 421220021$ T^16 - 22T^15 - 11T^14 + 3284T^13 - 17991T^12 - 98360T^11 + 848902T^10 + 1097628T^9 - 15757282T^8 - 6661432T^7 + 142379102T^6 + 63510920T^5 - 606143211T^4 - 508201966T^3 + 867909209T^2 + 1255079188*T + 421220021
$41$	$T^{16} + \cdots + 48434041616$ T^16 - 20T^15 - 116T^14 + 4716T^13 - 11516T^12 - 378692T^11 + 2300234T^10 + 10593764T^9 - 119337844T^8 + 55801684T^7 + 2265327436T^6 - 6229072796T^5 - 9045582139T^4 + 54361060624T^3 - 40930758904T^2 - 56552398912*T + 48434041616
$43$	$T^{16} + 26 T^{15} + \cdots + 10244096$ T^16 + 26T^15 + 89T^14 - 2848T^13 - 23884T^12 + 70848T^11 + 1263969T^10 + 1490382T^9 - 23114659T^8 - 70707152T^7 + 101879016T^6 + 613653184T^5 + 306996304T^4 - 1364579968T^3 - 1796408064T^2 - 521891840*T + 10244096
$47$	$T^{16} + \cdots - 2949551104$ T^16 + 24T^15 - 100T^14 - 6100T^13 - 19880T^12 + 498932T^11 + 2987098T^10 - 14266400T^9 - 116214500T^8 + 142576460T^7 + 1762475608T^6 - 279072268T^5 - 11200440375T^4 - 3143549400T^3 + 24625685920T^2 + 14478883584*T - 2949551104
$53$	$T^{16} - 16 T^{15} + \cdots + 91170496$ T^16 - 16T^15 - 200T^14 + 3960T^13 + 8160T^12 - 287228T^11 - 181322T^10 + 9372700T^9 + 9225280T^8 - 131297720T^7 - 244260672T^6 + 302059092T^5 + 589505125T^4 - 230774740T^3 - 440028440T^2 + 57533024*T + 91170496
$59$	$T^{16} + \cdots - 2071241191424$ T^16 - 40T^15 + 414T^14 + 4156T^13 - 119601T^12 + 686668T^11 + 3870514T^10 - 60858036T^9 + 137937061T^8 + 1377429904T^7 - 8286677044T^6 - 714461056T^5 + 120754417616T^4 - 272689616896T^3 - 364922395904T^2 + 2046353911808*T - 2071241191424
$61$	$T^{16} + \cdots - 57875079875$ T^16 - 22T^15 - 131T^14 + 5172T^13 - 899T^12 - 478360T^11 + 847390T^10 + 22322260T^9 - 48944690T^8 - 547707000T^7 + 1221648150T^6 + 6479855400T^5 - 15415376575T^4 - 28355642750T^3 + 83043567125T^2 - 7494103500*T - 57875079875
$67$	$T^{16} + \cdots + 10874045696$ T^16 + 50T^15 + 801T^14 - 228T^13 - 163176T^12 - 2129136T^11 - 10707539T^10 + 2219098T^9 + 269895541T^8 + 1085780072T^7 + 494097824T^6 - 7302497632T^5 - 18904593824T^4 - 10280143488T^3 + 20323074304T^2 + 29673819136*T + 10874045696
$71$	$T^{16} + \cdots - 17821854244720$ T^16 - 4T^15 - 636T^14 + 1580T^13 + 156580T^12 - 99824T^11 - 19363514T^10 - 22054996T^9 + 1254739440T^8 + 3294894620T^7 - 40160638516T^6 - 155777419296T^5 + 518568658161T^4 + 2822981253800T^3 - 922242954660T^2 - 17219048617280*T - 17821854244720
$73$	$T^{16} + \cdots + 467056954256$ T^16 - 32T^15 - 44T^14 + 11424T^13 - 98884T^12 - 884856T^11 + 15766166T^10 - 29863336T^9 - 593066164T^8 + 3397418304T^7 + 864800524T^6 - 47152069112T^5 + 76840707829T^4 + 178207450776T^3 - 454555238456T^2 - 95802961120*T + 467056954256
$79$	$T^{16} + \cdots - 39769573120$ T^16 - 8T^15 - 634T^14 + 3260T^13 + 161035T^12 - 340208T^11 - 20330526T^10 - 14950128T^9 + 1238575405T^8 + 3971808260T^7 - 26242593984T^6 - 152472338288T^5 - 137333541344T^4 + 373105565120T^3 + 506256640000T^2 - 109157228800*T - 39769573120
$83$	$T^{16} + \cdots - 5160481904$ T^16 + 30T^15 + 19T^14 - 7796T^13 - 74561T^12 + 287382T^11 + 7078254T^10 + 21667486T^9 - 140788269T^8 - 982733184T^7 - 859176229T^6 + 8139152086T^5 + 28274673621T^4 + 35701019196T^3 + 13562771676T^2 - 7223334928*T - 5160481904
$89$	$T^{16} + \cdots - 47559207352064$ T^16 - 28T^15 - 366T^14 + 15616T^13 + 139T^12 - 3056440T^11 + 12917242T^10 + 255199692T^9 - 1622477287T^8 - 9198149448T^7 + 73507624792T^6 + 149910524000T^5 - 1495732572176T^4 - 1079154290304T^3 + 13879308403584T^2 + 2732998940672*T - 47559207352064
$97$	$T^{16} + \cdots - 42316001699$ T^16 - 36T^15 + 140T^14 + 7740T^13 - 70810T^12 - 549288T^11 + 7342748T^10 + 12044620T^9 - 310784805T^8 + 112539320T^7 + 5661565548T^6 - 4262188388T^5 - 46130666510T^4 + 23281040940T^3 + 131521397240T^2 - 37993251136*T - 42316001699
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 $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.16
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more