
// Make newform 1815.2.c.e in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1815_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1815_c_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_1815_2_c_e();".
// 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_1815_2_c_e();".
// 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
        poly := [2, -4, 4, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [38, 2, -1, -4, 8, -1], [29, 10, -5, -20, 17, -5], [-13, 32, 30, 5, -10, 7], [27, -70, -11, -21, 19, -11], [26, -64, -37, -10, 20, -14]];
    Rf_basisdens := [1, 23, 23, 23, 23, 23];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1815_c();"
function MakeCharacter_1815_c()
    N := 1815;
    order := 2;
    char_gens := [1211, 727, 1696];
    v := [2, 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_1815_c_Hecke();"
function MakeCharacter_1815_c_Hecke(Kf)
    N := 1815;
    order := 2;
    char_gens := [1211, 727, 1696];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+1)]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0], [0, 0, -1, 0, 0, 1], [0, 0, 0, -1, -1, 1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, 1], [0, 0, 0, -1, 3, -1], [-2, -2, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0], [2, -2, -2, 0, 0, 0], [-2, 0, 2, 0, 0, 0], [0, 0, 0, 2, -2, 2], [-2, -6, 2, 0, 0, 0], [0, 0, 0, 5, -3, 3], [0, 0, 0, 6, -2, -2], [0, 0, 0, -4, -4, 2], [-4, -2, -4, 0, 0, 0], [6, 0, -4, 0, 0, 0], [0, 0, 0, 4, 0, 4], [-4, -6, 0, 0, 0, 0], [0, 0, 0, -5, 5, -1], [-2, 6, 4, 0, 0, 0], [0, 0, 0, -3, -1, 5], [-8, 0, -6, 0, 0, 0], [0, 0, 0, 4, 4, -8], [-6, 6, -10, 0, 0, 0], [0, 0, 0, 2, 6, 6], [0, 0, 0, -1, -3, 7], [-10, 0, 0, 0, 0, 0], [0, 0, 0, -6, 2, -4], [0, 0, 0, 1, -3, -5], [0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, -2], [-6, -2, -8, 0, 0, 0], [6, 2, -2, 0, 0, 0], [6, -2, 0, 0, 0, 0], [0, 0, 0, 2, -2, 2], [0, 0, 0, -6, -2, -2], [0, 0, 0, -7, -1, -3], [0, 0, 0, -15, 9, -3], [4, 4, 0, 0, 0, 0], [-6, 6, -8, 0, 0, 0], [8, -4, 0, 0, 0, 0], [0, 0, 0, -5, 1, -9], [0, 0, 0, -1, -9, -5], [-4, -8, 8, 0, 0, 0], [10, -2, 8, 0, 0, 0], [0, 0, 0, -18, 2, -2], [0, 0, 0, 5, -5, -3], [-2, 4, -12, 0, 0, 0], [0, 0, 0, -17, -1, -1], [-8, 8, -12, 0, 0, 0], [6, 4, 8, 0, 0, 0], [-8, 6, 8, 0, 0, 0], [0, 0, 0, -10, 6, 0], [0, 0, 0, 21, -1, 1], [2, 4, 16, 0, 0, 0], [6, 10, -4, 0, 0, 0], [0, 0, 0, 3, 1, 11], [-6, 2, -2, 0, 0, 0], [0, 0, 0, -7, -3, -5], [0, 0, 0, -13, 7, -9], [0, 0, 0, -1, 11, 5], [-8, 6, -12, 0, 0, 0], [0, 0, 0, 2, -10, -6], [0, 0, 0, 6, -10, 12], [-2, 0, 2, 0, 0, 0], [0, 0, 0, -7, 3, 5], [0, 0, 0, 7, 1, 7], [6, -8, 8, 0, 0, 0], [0, 0, 0, -4, 4, -2], [-4, 4, 4, 0, 0, 0], [0, 0, 0, 10, -2, -6], [0, 0, 0, 11, -15, 3], [-12, 12, -8, 0, 0, 0], [0, 0, 0, -14, -6, -2], [-14, 8, -4, 0, 0, 0], [0, 0, 0, 0, 16, 0], [0, 0, 6, 0, 0, 0], [-10, -4, 8, 0, 0, 0], [4, 4, 0, 0, 0, 0], [-2, -10, -8, 0, 0, 0], [0, 0, 4, 0, 0, 0], [0, 0, 0, 16, 0, 0], [2, 2, 0, 0, 0, 0], [0, 0, 0, -26, -2, -2], [24, -8, 14, 0, 0, 0], [0, 0, 0, 7, -15, -5], [-14, 10, -14, 0, 0, 0], [0, 0, 0, -26, 2, 6], [0, 0, 0, -4, -8, -4], [16, 4, 4, 0, 0, 0], [0, 0, 0, 20, 8, 4], [8, 0, 0, 0, 0, 0], [-24, -4, 0, 0, 0, 0], [0, 0, 0, -5, -7, 15], [-2, -12, 0, 0, 0, 0], [-8, 0, -10, 0, 0, 0], [0, 0, 0, -1, 11, -3], [6, -4, 4, 0, 0, 0], [0, 0, 0, 17, 5, -5], [0, 0, 0, 7, 3, 11], [0, 0, 0, 7, -11, -5], [-18, -2, -2, 0, 0, 0], [2, -6, 0, 0, 0, 0], [0, 0, 0, -22, 22, -6], [0, 0, 0, -8, -4, 8], [0, 0, 0, -19, -3, -3], [24, 8, -8, 0, 0, 0], [14, -12, -4, 0, 0, 0], [0, 0, 0, 9, -11, 11], [0, 0, 0, -9, 9, -9], [0, 0, 0, 22, -2, -8], [-6, 16, 6, 0, 0, 0], [4, 0, -16, 0, 0, 0], [-4, -4, -14, 0, 0, 0], [0, 0, 0, 14, 2, -14], [0, 0, 0, -6, 2, -2], [0, 0, 0, 2, -14, 16], [24, -4, -4, 0, 0, 0], [-18, -4, -12, 0, 0, 0], [0, 0, 0, -21, -3, -1], [0, 0, 0, 25, -3, 1], [0, 0, 0, 14, -10, 2], [20, 0, 16, 0, 0, 0], [6, -6, -2, 0, 0, 0], [-6, -6, -20, 0, 0, 0], [-8, -12, 8, 0, 0, 0], [0, 0, 0, 16, -4, 4], [0, 0, 0, -37, 1, 3], [-6, 10, -12, 0, 0, 0], [0, 0, 0, -11, 7, -11], [4, 12, 0, 0, 0, 0], [0, 0, 0, -10, 2, 2], [-2, 6, -2, 0, 0, 0], [-18, 12, 0, 0, 0, 0], [0, 0, 0, -6, 18, 8], [0, 0, 0, 27, -5, 5], [0, 0, 0, -18, -10, 16], [18, -10, 26, 0, 0, 0], [18, 6, -12, 0, 0, 0], [-26, -2, -14, 0, 0, 0], [0, 0, 0, -10, 2, 10], [0, 0, 0, 3, -3, 3], [14, 2, 4, 0, 0, 0], [8, 10, -4, 0, 0, 0], [0, 0, 0, -5, 1, 11], [0, 0, 0, 23, 7, 3], [2, 16, -18, 0, 0, 0], [0, 0, 0, -6, 18, -2], [0, 0, 0, 5, 19, 5], [-2, 4, 4, 0, 0, 0], [0, 0, 0, -4, -16, 8], [0, 0, 0, -23, 3, 1], [0, 0, 0, -34, 18, -2], [4, 12, -20, 0, 0, 0], [-6, -14, 16, 0, 0, 0], [-2, -20, -4, 0, 0, 0], [0, 0, 0, -1, 13, -21], [-18, -18, -6, 0, 0, 0], [0, 0, 0, 18, -6, -2], [0, 0, 0, 7, 19, 15], [0, 0, 0, -15, 29, -5], [-24, 16, -4, 0, 0, 0], [0, 0, 0, -8, 16, -22], [0, 0, 0, 10, 10, 6], [-6, 8, -2, 0, 0, 0], [0, 0, 0, 13, -5, -11], [-2, -20, -4, 0, 0, 0], [0, 0, 0, -20, 12, -10], [0, 4, 12, 0, 0, 0], [-18, -6, -8, 0, 0, 0], [24, 20, -8, 0, 0, 0], [0, 0, 0, -15, 11, -19], [10, -12, -6, 0, 0, 0], [-6, 12, 4, 0, 0, 0], [18, -14, 8, 0, 0, 0], [-10, 20, -24, 0, 0, 0], [0, 0, 0, 35, 7, -7], [-34, -12, -4, 0, 0, 0], [0, 0, 0, 40, 4, -4], [-36, 4, -16, 0, 0, 0], [0, 0, 0, -26, 2, -14], [0, 0, 0, 17, -7, 21], [0, 0, 0, 10, 2, -30], [16, -16, 14, 0, 0, 0], [0, 0, 0, 13, 15, -11], [0, 0, 0, -2, 18, -26], [-26, -16, 0, 0, 0, 0], [0, 20, 4, 0, 0, 0], [0, 0, 0, 6, 26, 2], [0, 0, 0, 13, 3, -3], [24, -4, -8, 0, 0, 0], [18, 8, -4, 0, 0, 0], [0, 0, 0, 29, 7, -23], [0, 0, 0, -10, 14, 4], [46, -4, 4, 0, 0, 0], [0, 0, 0, 4, -12, 8], [0, 0, 0, -39, 21, -7], [0, 0, 0, 29, -21, 13], [-26, -6, -10, 0, 0, 0], [-14, -10, 16, 0, 0, 0], [0, 0, 0, 24, -8, -12], [-10, 20, -20, 0, 0, 0], [-8, -20, 4, 0, 0, 0], [0, 0, 0, 10, -14, 20], [-16, 0, 8, 0, 0, 0], [0, 0, 0, 23, -23, -5], [18, -18, 2, 0, 0, 0], [10, -20, -14, 0, 0, 0], [0, 0, 0, 3, -3, -17], [0, -24, -6, 0, 0, 0], [0, 0, 0, -8, -20, 4], [0, 0, 0, -16, 4, -8], [20, -20, 20, 0, 0, 0], [18, 8, 16, 0, 0, 0], [0, 0, 0, -45, -5, -3], [-18, 18, 2, 0, 0, 0], [0, 0, 0, 32, -20, 16], [0, 0, 0, 10, -14, 8], [-18, -16, 0, 0, 0, 0], [6, -10, 12, 0, 0, 0], [-30, 20, -20, 0, 0, 0], [0, 0, 0, 2, 14, -18], [0, 0, 0, 15, -31, -5], [-10, -4, -12, 0, 0, 0], [0, 0, 0, 40, -16, -6], [8, -14, 4, 0, 0, 0], [0, 0, 0, 5, -3, 11], [-4, -20, -4, 0, 0, 0], [0, 0, 0, -8, 16, -4], [14, 30, -12, 0, 0, 0], [-14, -2, 0, 0, 0, 0], [-26, -18, -2, 0, 0, 0], [0, 0, 0, 20, 24, 4], [0, 0, 0, -13, -15, -5], [14, 10, -12, 0, 0, 0], [0, 0, 0, 27, -9, -1], [48, -2, 0, 0, 0, 0], [20, 24, -12, 0, 0, 0], [0, 0, 0, 6, 14, -2], [-22, 2, -4, 0, 0, 0], [0, 0, 0, 16, -36, 28], [-18, 6, -28, 0, 0, 0], [0, 0, 0, -1, 23, -9], [-12, 20, -4, 0, 0, 0], [0, 0, 0, 16, -4, -16], [-40, -6, 8, 0, 0, 0], [14, 18, -8, 0, 0, 0], [0, 0, 0, -11, 19, -27], [0, 0, 0, 7, 29, -17], [10, 18, -2, 0, 0, 0], [0, 0, 0, 8, -4, -8], [-2, -20, -4, 0, 0, 0], [0, 0, 0, 21, -31, 1], [36, 8, -4, 0, 0, 0], [-6, 6, 12, 0, 0, 0], [0, 0, 0, 25, -3, 11], [0, 0, 0, 18, 2, 24], [0, 0, 0, -21, 9, -9], [0, 0, 0, 21, -3, 27], [0, 0, 0, 9, 3, -11], [-2, 24, -12, 0, 0, 0], [0, 0, 0, -47, 11, -7], [0, 0, 0, 0, 16, -30], [-48, 8, 8, 0, 0, 0], [-4, 28, -14, 0, 0, 0], [-22, 12, -12, 0, 0, 0], [0, 0, 0, 29, 5, -5], [0, 0, 0, -9, 3, -1], [10, -8, -8, 0, 0, 0], [0, 0, 0, -30, -10, 18], [0, 0, 0, -50, 2, 10], [-10, -6, -28, 0, 0, 0], [0, 0, 0, 3, 5, -1], [0, 0, 0, 28, 0, -12], [0, 0, 0, 6, -18, 26], [-26, -24, 16, 0, 0, 0], [-2, 8, 4, 0, 0, 0], [-8, 16, -4, 0, 0, 0], [0, 0, 0, 1, -1, -19], [18, -8, 38, 0, 0, 0], [0, 0, 0, -25, 25, 7], [6, -8, -8, 0, 0, 0], [0, 0, 0, 17, 17, -41], [-32, 8, 16, 0, 0, 0], [0, 0, 0, 28, 20, 8], [0, 0, 0, -27, -3, -7], [-42, 8, 10, 0, 0, 0], [14, 18, 2, 0, 0, 0], [-6, -8, 20, 0, 0, 0], [0, 0, 0, -30, 10, -22], [0, 0, 0, -13, -17, 23], [24, 12, -12, 0, 0, 0], [0, 0, 0, 1, 11, 25], [18, 10, -14, 0, 0, 0], [-12, 16, 0, 0, 0, 0], [0, 0, 0, -46, 2, -4], [12, -20, -36, 0, 0, 0], [0, 0, 0, -29, -13, 5], [0, 0, 0, 1, -25, 13], [0, 0, 0, 11, 11, 3], [42, -18, 20, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1815_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1815_2_c_e();".
// 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_1815_2_c_e(:prec:=6)
    chi := MakeCharacter_1815_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1815_2_c_e();".
// 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_1815_2_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1815_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, 11, 0, 7, 0, 1]>,<19,R![-40, -4, 6, 1]>],Snew);
    return Vf;
end function;