[N,k,chi] = [192,9,Mod(127,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.127");
S:= CuspForms(chi, 9);
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 \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).
\(n\)
\(65\)
\(127\)
\(133\)
\(\chi(n)\)
\(1\)
\(-1\)
\(1\)
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
\( T_{5}^{4} + 280T_{5}^{3} - 1356168T_{5}^{2} - 444757280T_{5} + 95173421200 \)
T5^4 + 280*T5^3 - 1356168*T5^2 - 444757280*T5 + 95173421200
acting on \(S_{9}^{\mathrm{new}}(192, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{8} \)
T^8
$3$
\( (T^{2} + 2187)^{4} \)
(T^2 + 2187)^4
$5$
\( (T^{4} + 280 T^{3} + \cdots + 95173421200)^{2} \)
(T^4 + 280*T^3 - 1356168*T^2 - 444757280*T + 95173421200)^2
$7$
\( T^{8} + 27860224 T^{6} + \cdots + 50\!\cdots\!84 \)
T^8 + 27860224*T^6 + 208130158086144*T^4 + 281337835843759636480*T^2 + 501546202243916008259584
$11$
\( T^{8} + 649524160 T^{6} + \cdots + 89\!\cdots\!56 \)
T^8 + 649524160*T^6 + 103705089250223616*T^4 + 1868162841608529176412160*T^2 + 8925321456605961181081758662656
$13$
\( (T^{4} - 27128 T^{3} + \cdots + 78\!\cdots\!96)^{2} \)
(T^4 - 27128*T^3 - 1547259240*T^2 + 43945280225824*T + 7812656908270096)^2
$17$
\( (T^{4} - 218088 T^{3} + \cdots + 69\!\cdots\!44)^{2} \)
(T^4 - 218088*T^3 + 17181588312*T^2 - 578424127540896*T + 6998294019876163344)^2
$19$
\( T^{8} + 83516612544 T^{6} + \cdots + 23\!\cdots\!64 \)
T^8 + 83516612544*T^6 + 1687644177657349252608*T^4 + 11208089478557060075425381466112*T^2 + 23614274032518286815711674239926904356864
$23$
\( T^{8} + 249703618816 T^{6} + \cdots + 21\!\cdots\!64 \)
T^8 + 249703618816*T^6 + 17195342302371079938048*T^4 + 404569291765506002304505266307072*T^2 + 2127998919130293407017231185339602975064064
$29$
\( (T^{4} + 97976 T^{3} + \cdots + 81\!\cdots\!44)^{2} \)
(T^4 + 97976*T^3 - 704307414216*T^2 - 82910803678994848*T + 81354638075299384633744)^2
$31$
\( T^{8} + 1943611464448 T^{6} + \cdots + 24\!\cdots\!96 \)
T^8 + 1943611464448*T^6 + 1202674618071716419405824*T^4 + 293605469331260832124273796282318848*T^2 + 24726551271311810899552438533139853747128631296
$37$
\( (T^{4} - 3511704 T^{3} + \cdots + 36\!\cdots\!28)^{2} \)
(T^4 - 3511704*T^3 - 147261364776*T^2 + 1649236071201559200*T + 360760332501783549178128)^2
$41$
\( (T^{4} + 6017560 T^{3} + \cdots - 53\!\cdots\!16)^{2} \)
(T^4 + 6017560*T^3 - 332686454952*T^2 - 47237488456003970720*T - 53078528977164103900184816)^2
$43$
\( T^{8} + 18578691622336 T^{6} + \cdots + 10\!\cdots\!00 \)
T^8 + 18578691622336*T^6 + 113735689709191707439597056*T^4 + 250306426155420197568253956154700185600*T^2 + 105221482520071034219340064207437422764336783360000
$47$
\( T^{8} + 104203981431808 T^{6} + \cdots + 23\!\cdots\!76 \)
T^8 + 104203981431808*T^6 + 2681432654872175543143661568*T^4 + 5052649139712016671282280440610064171008*T^2 + 2359295550475787720937544681894223764989627099774976
$53$
\( (T^{4} + 13671096 T^{3} + \cdots - 44\!\cdots\!24)^{2} \)
(T^4 + 13671096*T^3 - 196100030026440*T^2 - 2975809927461327518112*T - 4438747556765498496228634224)^2
$59$
\( T^{8} + 685106367582400 T^{6} + \cdots + 31\!\cdots\!04 \)
T^8 + 685106367582400*T^6 + 112742352757450615499258279424*T^4 + 401041916339167960927382116720848246587392*T^2 + 316462066724579934994122574583203448332118019007381504
$61$
\( (T^{4} + 25401640 T^{3} + \cdots - 27\!\cdots\!32)^{2} \)
(T^4 + 25401640*T^3 + 167853967511640*T^2 - 130471684111591285856*T - 2734694894421963774790971632)^2
$67$
\( T^{8} + \cdots + 22\!\cdots\!56 \)
T^8 + 1663125039605440*T^6 + 995395621133540461425148368384*T^4 + 251424786571249510300686566726474034950225920*T^2 + 22286355769980050863191002824291065151027058020198326009856
$71$
\( T^{8} + \cdots + 33\!\cdots\!76 \)
T^8 + 4281150145613056*T^6 + 5434388145540632047651100123136*T^4 + 2191194198380716151204137994426423623463993344*T^2 + 33285810780568379427083024480127147988214149978832163045376
$73$
\( (T^{4} - 29770824 T^{3} + \cdots - 16\!\cdots\!68)^{2} \)
(T^4 - 29770824*T^3 - 1529010354731112*T^2 + 47361454689843607426272*T - 160605728523635947876144489968)^2
$79$
\( T^{8} + \cdots + 31\!\cdots\!84 \)
T^8 + 10818975487992064*T^6 + 41298140013813660691665424140288*T^4 + 64011999013959902270706277373119216527687811072*T^2 + 31972296484212060978754629048530752985944597250872939704745984
$83$
\( T^{8} + \cdots + 15\!\cdots\!24 \)
T^8 + 9505987207399360*T^6 + 28628010297251183583381365270016*T^4 + 27918667916745255581706317706258054674083201024*T^2 + 1500689400977496452348324249866624356599249044794141875175424
$89$
\( (T^{4} + 85397496 T^{3} + \cdots - 47\!\cdots\!32)^{2} \)
(T^4 + 85397496*T^3 - 5859630056044008*T^2 - 374858419924333236250656*T - 4772222719992389274006120552432)^2
$97$
\( (T^{4} - 136323592 T^{3} + \cdots - 88\!\cdots\!36)^{2} \)
(T^4 - 136323592*T^3 - 229470833760744*T^2 + 225752503781020603580384*T - 882173438862522427101749348336)^2
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