gp: [N,k,chi] = [368,3,Mod(45,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 3, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.45");
S:= CuspForms(chi, 3);
N := Newforms(S);
Newform invariants
sage: traces = [176]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
The algebraic q q q -expansion of this newform has not been computed, but we have computed the trace expansion .
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
T 3 88 + 2 T 3 87 + 2 T 3 86 − 46 T 3 85 + 4983 T 3 84 + 9104 T 3 83 + ⋯ + 13 ⋯ 64 T_{3}^{88} + 2 T_{3}^{87} + 2 T_{3}^{86} - 46 T_{3}^{85} + 4983 T_{3}^{84} + 9104 T_{3}^{83} + \cdots + 13\!\cdots\!64 T 3 8 8 + 2 T 3 8 7 + 2 T 3 8 6 − 4 6 T 3 8 5 + 4 9 8 3 T 3 8 4 + 9 1 0 4 T 3 8 3 + ⋯ + 1 3 ⋯ 6 4
T3^88 + 2*T3^87 + 2*T3^86 - 46*T3^85 + 4983*T3^84 + 9104*T3^83 + 9300*T3^82 - 214364*T3^81 + 10916705*T3^80 + 17750830*T3^79 + 18628502*T3^78 - 429687738*T3^77 + 13919571427*T3^76 + 19541934156*T3^75 + 21238698008*T3^74 - 489366886552*T3^73 + 11522932670113*T3^72 + 13486552350498*T3^71 + 15357142440594*T3^70 - 352657296871902*T3^69 + 6537781562967739*T3^68 + 6123298326228872*T3^67 + 7433532700557484*T3^66 - 169616110260572164*T3^65 + 2614965445104396365*T3^64 + 1864063004918960670*T3^63 + 2479512934066511670*T3^62 - 55994284415092612698*T3^61 + 747093560275774893879*T3^60 + 378775345674525175796*T3^59 + 578274397924797530944*T3^58 - 12856421055888073770608*T3^57 + 152944519003002959734055*T3^56 + 49511710596604492300118*T3^55 + 94743097921253282088086*T3^54 - 2058835960111344085012858*T3^53 + 22341164947628011628776517*T3^52 + 3684132073803071421927936*T3^51 + 10875196282259850085230316*T3^50 - 228687277393350252575837092*T3^49 + 2304764553975086033338736275*T3^48 + 70784326127834575845485338*T3^47 + 867273641217544628219416626*T3^46 - 17395139134892362339655633598*T3^45 + 165310271309416473393894248473*T3^44 - 13117778212972323944496421388*T3^43 + 47463936553162949377771909304*T3^42 - 888983256277355927119209639896*T3^41 + 8073537817720678434864784736411*T3^40 - 1341198138623607020062476758138*T3^39 + 1757293015266587573300067162550*T3^38 - 29788726807613455679337365093402*T3^37 + 261544936527785446382142630123353*T3^36 - 60454886899158263631991371655768*T3^35 + 43348704197861044004723062296724*T3^34 - 635627661600150425859259124809084*T3^33 + 5441286455662561995654284144467647*T3^32 - 1507809345119900817694167655760742*T3^31 + 701132905047664203429509188071170*T3^30 - 8355285062136378990499423266352430*T3^29 + 69853100484212234620447520268335501*T3^28 - 21629655654579408970182809222173876*T3^27 + 7261134897779672065658194027899248*T3^26 - 64542214963017243883167404778604448*T3^25 + 525715465438292793882636624646510172*T3^24 - 175735986274146259940649088176947968*T3^23 + 46134472762341019247355900463803680*T3^22 - 271135065343073655927985038202430048*T3^21 + 2182197282775750248653580073948220272*T3^20 - 756579539093318129895123731360516160*T3^19 + 167486642720160714149636928694751744*T3^18 - 579202554487923495913372674984423168*T3^17 + 4651674610128621340921650056868777024*T3^16 - 1587875210471480090438455126556440576*T3^15 + 316917559435613042837829922667347968*T3^14 - 653467687007709635119193188352106496*T3^13 + 4497014371435948366842095089007136768*T3^12 - 1508730726427763069049432771102507008*T3^11 + 291837473316900798175124923119042560*T3^10 - 406639907518362186921395786301046784*T3^9 + 1380112894855377954749605022349754368*T3^8 - 572508924255994279838262977744076800*T3^7 + 124327855702597404440804057623298048*T3^6 - 32915639545622850027211487633735680*T3^5 + 76937645284387260050842818204139520*T3^4 - 35317822630394279929289140036698112*T3^3 + 8126378438523465888568967168000000*T3^2 - 47620874612438683692090523648000*T3 + 139530032720556443744509886464
acting on S 3 n e w ( 368 , [ χ ] ) S_{3}^{\mathrm{new}}(368, [\chi]) S 3 n e w ( 3 6 8 , [ χ ] ) .