[N,k,chi] = [144,6,Mod(49,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
\(n\)
\(37\)
\(65\)
\(127\)
\(\chi(n)\)
\(1\)
\(\beta_{1}\)
\(1\)
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.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} + 21 T_{5}^{9} + 10422 T_{5}^{8} - 323055 T_{5}^{7} + 91398294 T_{5}^{6} - 926555355 T_{5}^{5} + 74758828257 T_{5}^{4} - 900082524816 T_{5}^{3} + \cdots + 42\!\cdots\!24 \)
T5^10 + 21*T5^9 + 10422*T5^8 - 323055*T5^7 + 91398294*T5^6 - 926555355*T5^5 + 74758828257*T5^4 - 900082524816*T5^3 + 53123350023936*T5^2 - 457491118694400*T5 + 4234048679706624
acting on \(S_{6}^{\mathrm{new}}(144, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{10} \)
T^10
$3$
\( T^{10} + 12 T^{9} + \cdots + 847288609443 \)
T^10 + 12*T^9 + 66*T^8 - 2160*T^7 + 72981*T^6 + 1732104*T^5 + 17734383*T^4 - 127545840*T^3 + 947027862*T^2 + 41841412812*T + 847288609443
$5$
\( T^{10} + 21 T^{9} + \cdots + 42\!\cdots\!24 \)
T^10 + 21*T^9 + 10422*T^8 - 323055*T^7 + 91398294*T^6 - 926555355*T^5 + 74758828257*T^4 - 900082524816*T^3 + 53123350023936*T^2 - 457491118694400*T + 4234048679706624
$7$
\( T^{10} + 29 T^{9} + \cdots + 56\!\cdots\!04 \)
T^10 + 29*T^9 + 44466*T^8 - 2300235*T^7 + 1392244974*T^6 - 75213929775*T^5 + 22593265190433*T^4 - 1826372997396420*T^3 + 258259673848446996*T^2 - 11838881253749666512*T + 569976119817504713104
$11$
\( T^{10} + 177 T^{9} + \cdots + 17\!\cdots\!25 \)
T^10 + 177*T^9 + 409023*T^8 + 156952350*T^7 + 161733498741*T^6 + 43348856880519*T^5 + 12558885570039933*T^4 + 931807571894691318*T^3 + 150597355001578760799*T^2 + 973534255980143118345*T + 1798512850186689084921225
$13$
\( T^{10} + 181 T^{9} + \cdots + 14\!\cdots\!36 \)
T^10 + 181*T^9 + 844134*T^8 + 216094281*T^7 + 536032392354*T^6 + 128853638078853*T^5 + 151974583529204709*T^4 + 33136080731853487872*T^3 + 30923019224440692384156*T^2 + 5859616568225136928559392*T + 1426544527661232177553557136
$17$
\( (T^{5} - 1140 T^{4} + \cdots + 113704762586184)^{2} \)
(T^5 - 1140*T^4 - 2118735*T^3 + 3030697566*T^2 - 1066166319636*T + 113704762586184)^2
$19$
\( (T^{5} - 416 T^{4} + \cdots + 97\!\cdots\!92)^{2} \)
(T^5 - 416*T^4 - 7334237*T^3 - 2095532308*T^2 + 13442524247504*T + 9782364002768192)^2
$23$
\( T^{10} + 399 T^{9} + \cdots + 46\!\cdots\!96 \)
T^10 + 399*T^9 + 16104474*T^8 + 45813128391*T^7 + 256756407066486*T^6 + 431345646852301611*T^5 + 797951645471248867449*T^4 + 485139034341060407918100*T^3 + 628022413558447543950638580*T^2 + 171354315215004665183683275024*T + 469963179322401974659599617742096
$29$
\( T^{10} + 6033 T^{9} + \cdots + 96\!\cdots\!36 \)
T^10 + 6033*T^9 + 106207902*T^8 + 402872482485*T^7 + 6982404272464842*T^6 + 27319508978657387769*T^5 + 177348152345359399294005*T^4 + 278008210630976784402054600*T^3 + 1420641164736425459513326948956*T^2 + 1169517391017316313666204759168448*T + 9630324142326544231270176259487985936
$31$
\( T^{10} + 2759 T^{9} + \cdots + 69\!\cdots\!00 \)
T^10 + 2759*T^9 + 88243278*T^8 + 426408133899*T^7 + 6788262652151238*T^6 + 25438125963411614607*T^5 + 147031760681605937546361*T^4 + 227566045811818548648364080*T^3 + 1224794438981204298911185952400*T^2 + 1602625003499662952610784851680000*T + 6942401596532791478956698186436000000
$37$
\( (T^{5} + 7586 T^{4} + \cdots + 21\!\cdots\!00)^{2} \)
(T^5 + 7586*T^4 - 145518692*T^3 - 1301026358312*T^2 + 1428492776519168*T + 21920819628401968000)^2
$41$
\( T^{10} + 18435 T^{9} + \cdots + 85\!\cdots\!09 \)
T^10 + 18435*T^9 + 461601963*T^8 + 2062935126630*T^7 + 54390089528524629*T^6 + 266394833820322597737*T^5 + 4484858462794803405584925*T^4 + 1954731380997396006659627622*T^3 + 19927673901074661237075152069235*T^2 + 1271320289580540699751550644119555*T + 85449302117497270486410345728982072009
$43$
\( T^{10} + 1469 T^{9} + \cdots + 66\!\cdots\!21 \)
T^10 + 1469*T^9 + 463178679*T^8 - 3368564386026*T^7 + 163643432526985509*T^6 - 1015259270617240836213*T^5 + 23819427249554697569128629*T^4 - 173854293545304276433644401682*T^3 + 2547352333609824301280784491237151*T^2 - 12059836098867646818773793077845863787*T + 66064599810092446543354395683111361298321
$47$
\( T^{10} - 25155 T^{9} + \cdots + 18\!\cdots\!24 \)
T^10 - 25155*T^9 + 745478802*T^8 - 2958341092011*T^7 + 70002824199553998*T^6 + 461020322523830995521*T^5 + 10254741148031106265869681*T^4 + 46071535503232091309264764236*T^3 + 229717842491448021201090892192980*T^2 + 67683892303425445346513241134015472*T + 18905753095992229246237122912444315024
$53$
\( (T^{5} - 58422 T^{4} + \cdots - 86\!\cdots\!28)^{2} \)
(T^5 - 58422*T^4 + 1263590748*T^3 - 12585968591112*T^2 + 56566983606862464*T - 86411235527486540928)^2
$59$
\( T^{10} - 90537 T^{9} + \cdots + 48\!\cdots\!81 \)
T^10 - 90537*T^9 + 6405739335*T^8 - 207986454772038*T^7 + 5843326331426559669*T^6 - 58316831203247137134423*T^5 + 1729718624799701884701189405*T^4 - 20773654503633275553523520877582*T^3 + 263916406136845178469480842869629879*T^2 - 1240779820068187959745516524878897498097*T + 4893738186723530808421496636083483389192681
$61$
\( T^{10} - 1403 T^{9} + \cdots + 45\!\cdots\!24 \)
T^10 - 1403*T^9 + 686075622*T^8 - 4007745839055*T^7 + 416930581745459910*T^6 - 1613327278303620400443*T^5 + 43397510306867433918335313*T^4 + 43499966682767808818223804768*T^3 + 3143471699954006815917984748355904*T^2 - 3668726129643262742885386245911091200*T + 4522018079556937214919120939992728244224
$67$
\( T^{10} + 13907 T^{9} + \cdots + 20\!\cdots\!49 \)
T^10 + 13907*T^9 + 2873939991*T^8 + 133754061247050*T^7 + 9050009696855720325*T^6 + 246575846574309437882565*T^5 + 5522430156944153941323415989*T^4 + 50040108151222479248885318938866*T^3 + 333060179070779394320365265537617695*T^2 + 973086623437524120144322632451165861499*T + 2075365467031023230758052249646981692553649
$71$
\( (T^{5} + 114684 T^{4} + \cdots - 18\!\cdots\!64)^{2} \)
(T^5 + 114684*T^4 - 1262766096*T^3 - 401418831672000*T^2 - 6456975319152617472*T - 18055912642204050468864)^2
$73$
\( (T^{5} - 7600 T^{4} + \cdots - 17\!\cdots\!80)^{2} \)
(T^5 - 7600*T^4 - 6758523719*T^3 + 31964902867486*T^2 + 11344870273180522892*T - 17024581541505279868280)^2
$79$
\( T^{10} + 29993 T^{9} + \cdots + 13\!\cdots\!00 \)
T^10 + 29993*T^9 + 10032930366*T^8 + 635783520754389*T^7 + 92009757017835850806*T^6 + 4213210978586726275409457*T^5 + 242178247551430924353216603081*T^4 + 4311435976392042179605276530371040*T^3 + 190080902037935126968976280618388478544*T^2 + 1827438210198344501775404036851264525552640*T + 130899966228765882657117139431297705363839238400
$83$
\( T^{10} - 228951 T^{9} + \cdots + 16\!\cdots\!64 \)
T^10 - 228951*T^9 + 41098635198*T^8 - 3907824185416635*T^7 + 343614253426981753302*T^6 - 20928998993680490327028423*T^5 + 1464028259990356780974691409745*T^4 - 72051363164468153422446123853822680*T^3 + 3345784870247726090018083821273992382576*T^2 - 84169166033397583253471499238570183971858816*T + 1686611898478502215551433481262920935464795128064
$89$
\( (T^{5} - 299166 T^{4} + \cdots - 18\!\cdots\!72)^{2} \)
(T^5 - 299166*T^4 + 24695946936*T^3 + 21981512500848*T^2 - 48110869394595006192*T - 184619183890065347323872)^2
$97$
\( T^{10} - 40541 T^{9} + \cdots + 17\!\cdots\!25 \)
T^10 - 40541*T^9 + 25043205147*T^8 + 1218915608584182*T^7 + 456613879724763313077*T^6 + 11408434618897083967422105*T^5 + 2071195218545745399544625373405*T^4 + 50125801624973083490645466807733302*T^3 + 7479853889164458194005501505186508521091*T^2 + 112608560780124386310839182980854254067176675*T + 1736659692148417508083114064402202841675997955625
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