
// Make newform 36.17.d.a in Magma, downloaded from the LMFDB on 25 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_36_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_36_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_36_17_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_36_17_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_36_d();"
function MakeCharacter_36_d()
    N := 36;
    order := 2;
    char_gens := [19, 29];
    v := [1, 2];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_36_d_Hecke();"
function MakeCharacter_36_d_Hecke( : Kf := false)
    N := 36;
    order := 2;
    char_gens := [19, 29];
    char_values := [[-1], [1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 17;
    raw_aps := [[-256], [0], [-329666], [0], [0], [-1631232958], [9937278718], [0], [0], [-981515008322], [0], [-6167627357758], [3168324620158], [0], [0], [31962705295678], [0], [45990056420162], [0], [0], [1381042818437762], [0], [0], [6957151819021438], [14385701036152322], [517601979623998], [0], [0], [-1955125036762558], [-25756333160759042], [0], [0], [-189374321741850242], [0], [456937687754836798], [0], [-99780723719044798], [0], [0], [1228179424835490238], [0], [1535468476951039682], [0], [-2158764626596070398], [-1890796042333840322], [0], [0], [0], [0], [-7906344086917423678], [14355312433838363518], [0], [-11755228446496508158], [0], [-20606734999305523202], [0], [26502320619692176318], [0], [-66556735568192504638], [-10369068934825668482], [0], [32409281067417038398], [0], [0], [147815896251430343042], [68024403946742934718], [0], [-112287337413394133758], [0], [-162143823658852459198], [-351405429436165240322], [0], [0], [704220681974974253762], [0], [0], [637627286008172159038], [222973230919822816322], [-932250094079558304002], [-1135855045366016530558], [0], [1681094266207581059522], [0], [-2266689013498682277118], [0], [0], [-2062946911069754035202], [-3772911917567783740798], [-416775643987467831362], [0], [0], [0], [0], [0], [0], [0], [8175460438682131796158], [2066639384753377591678], [0], [9931291563428278712642], [0], [13737924314572855089598], [0], [21524730161441206984318], [0], [19316498178726545679362], [0], [-18346806481926463315202], [0], [-33686899644458771356798], [0], [35783792812988628719042], [-8456906833932780209282], [0], [0], [46945952823816302684158], [0], [0], [41850480140962363423678], [0], [-59016683627215093803838], [13391750037818562869762], [-72080197710373680884162], [0], [0], [116016772073302000555198], [-126270121075326123908158], [0], [0], [63022979501966899778882], [0], [0], [0], [125229659119106656363202], [-206301488565562309119362], [156381112389535886254082], [115319989811982121035838], [0], [-323241550099520211405122], [348988851419725245144958], [0], [127443971421025782876478], [0], [0], [366210830804172019110722], [0], [-203143890268005259702078], [339197021569798318185598], [0], [0], [-694623638212535603336638], [689947607047303845515518], [0], [0], [0], [0], [0], [-488933658854428390033922], [-329951837716056425842558], [-694828125994815450431042], [0], [-1065271722822319795009922], [0], [0], [770187763251745273731838], [0], [0], [-1945837395859457426018878]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_36_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_36_17_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_36_17_d_a(:prec:=1)
    chi := MakeCharacter_36_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 17));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_36_17_d_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_36_17_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_36_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,17,sign)));
    Vf := Kernel([<5,R![329666, 1]>],Snew);
    return Vf;
end function;