
// Make newform 6762.2.a.ca in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6762_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_ca();".
// 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_6762_2_a_ca();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_6762_a();"
function MakeCharacter_6762_a()
    N := 6762;
    order := 1;
    char_gens := [2255, 3727, 3823];
    v := [1, 1, 1];
    // 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;

function MakeCharacter_6762_a_Hecke(Kf)
    return MakeCharacter_6762_a();
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 := [[1, 0], [-1, 0], [1, 0], [0, 0], [-3, 0], [0, -1], [0, 1], [2, 1], [-1, 0], [-5, 3], [1, -4], [-2, -5], [-6, 3], [-2, -4], [-2, 8], [-5, 2], [5, -3], [6, 0], [-10, 4], [-4, 7], [0, -4], [1, -1], [-1, -8], [-2, -9], [5, -1], [-4, 6], [8, 4], [1, -6], [-6, 8], [-16, -1], [-3, -2], [-3, 3], [-2, -12], [-6, 8], [8, 4], [-7, 4], [-6, 7], [0, -1], [-10, -6], [-2, -2], [16, 0], [4, 6], [0, 1], [3, -16], [-6, 2], [6, -8], [2, 2], [13, -2], [5, 2], [-10, -9], [-6, 15], [-12, -8], [17, -5], [-23, -2], [0, 1], [-6, 12], [-3, 1], [7, 0], [0, -2], [-14, -9], [10, 10], [-1, 16], [-16, -7], [-2, -1], [-21, 3], [17, -5], [16, 7], [-21, 5], [8, -16], [-2, -17], [10, 9], [2, 0], [5, 21], [-20, 4], [12, 12], [14, -9], [-4, 2], [-16, 11], [8, -6], [19, -4], [6, 0], [6, 18], [12, -4], [-30, 0], [3, -10], [23, -9], [-6, 3], [-1, 15], [-14, -12], [6, -14], [-26, -2], [2, -23], [-21, -6], [-19, 7], [6, -14], [4, 2], [-29, -9], [-26, 0], [-12, 20], [-10, -21], [20, 3], [-29, -8], [-1, 10], [-20, 10], [8, 23], [-15, 16], [1, 1], [8, -12], [-8, 5], [7, 24], [9, -20], [-16, 0], [2, -24], [0, -7], [1, 1], [10, 21], [12, -1], [-2, 2], [33, 9], [-10, -24], [6, -12], [27, -12], [37, 4], [17, -17], [48, -1], [9, -6], [-8, 1], [-14, 10], [21, -11], [0, -11], [18, 2], [-2, -3], [3, 23], [32, 5], [-6, 1], [1, 29], [-20, -2], [2, -18], [-13, 12], [18, 23], [14, 20], [19, -3], [24, 16], [1, 14], [36, -6], [-12, 21], [14, 10], [0, -19], [-18, -1], [12, 19], [28, 7], [6, -10], [24, 21], [40, 4], [-18, -5], [-10, 4], [8, 20], [12, -9], [-9, -27], [-27, -6], [22, -20], [-18, -14], [-19, -12], [13, -12], [18, 12], [-8, 1], [1, -26], [18, 19], [-9, -13], [-23, 1], [-7, -24], [30, -12], [40, -11], [3, 25], [-7, 20], [-12, 21], [46, -8], [3, -29], [-16, 0], [44, 3], [11, 10], [27, -12], [22, 1], [-28, 7], [14, 1], [15, -10], [-6, 3], [-2, -32], [-1, 24], [-14, 21], [-7, 4], [-15, -5], [14, -1], [18, 8], [-29, 22], [-2, -30], [-16, 2], [-16, -35], [-30, 11], [0, -5], [-27, 8], [30, 24], [26, 28], [30, -18], [32, -4], [23, 11], [36, 12], [1, 3], [22, 25], [-42, 7], [40, -2], [27, -9], [-7, -19], [33, 10], [-20, 10], [46, -12], [-9, -24], [18, 13], [-6, -28], [32, 6], [44, 8], [-24, 24], [-34, -18], [-19, 29], [-23, -5], [-4, 33], [10, 3], [-42, 13], [19, 3], [18, 28], [-6, -46], [-48, -8], [1, -23], [2, -38], [32, 21], [-24, -26], [-13, 27], [-1, -18], [32, 0], [54, -15], [-14, 6], [4, -9], [-37, 2], [-10, -28], [-14, -23], [18, 35], [-27, -30], [-20, 20], [2, 9], [-4, 9], [-12, 15], [22, -11], [-2, 12], [42, -16], [-37, 25], [37, 5], [-4, 11], [-22, -27], [-15, -31], [11, -6], [-32, 12], [27, -20], [14, 31], [-14, 17], [12, 17], [-2, -32], [9, -6], [-44, 21], [30, -5], [-11, -37], [28, -26], [12, 3], [35, 21], [39, 11], [45, 10], [2, 2], [51, 15], [-68, 9], [-66, -6], [-18, -44], [-24, -15], [-37, -23], [-12, 0], [54, -11], [14, -14], [4, 46], [-37, 21], [-33, 8], [-45, -12], [-28, -40], [-42, 0], [-27, 8], [-6, 15], [-26, -10], [-40, 0], [-39, -8], [32, -10], [-58, 0], [-30, -8], [-26, 8], [42, -12], [-9, 22], [-11, -31], [-7, 39], [2, 32], [8, 28], [-35, 7], [14, -14], [10, -5], [-6, 23], [30, -26], [-16, 22], [16, 31], [-44, 5], [24, -29], [-18, -28], [-13, 3], [-6, -25], [33, 3], [-2, -25], [-34, 7], [-22, -42], [33, 19], [-66, 18], [-52, 20], [27, -25], [-48, -13], [38, 27], [2, -22], [-37, -4], [-22, 19], [-1, 13], [30, -26], [-68, 10], [28, -3], [38, 32], [8, 0], [-14, -56], [51, 22], [7, -5], [-54, -1], [-4, -44], [-25, 3], [19, 15], [-83, 2], [33, -9], [6, -34], [26, 19], [18, 26], [51, -4], [32, 15], [-51, 20], [-6, -4], [-27, 45], [-68, -1], [-34, 26], [-50, -9], [19, 36], [10, -15], [-12, -49], [14, -2], [-52, -6], [12, 43], [6, -14], [-8, 18], [-57, 24], [68, 6], [12, 16], [-45, 29], [23, 9], [-52, 18], [-86, 0], [-29, -16], [-12, 18], [-20, 42], [-30, -12], [-16, -8], [-3, 39], [-66, 18], [-24, -50], [12, -58], [-70, -2], [-26, 0], [28, 16], [36, 5], [86, 6], [-78, 6], [60, -15], [-2, -57], [-44, -19], [49, 13], [6, 26], [55, 21], [-2, 56], [-70, 1], [-11, 3], [11, -23], [-6, 2], [30, 43], [1, -37], [68, -6], [-68, -16], [16, -54], [-18, -13], [53, 21], [-15, 2], [-78, -4], [-88, -11], [58, -26], [37, 33], [19, 8], [-45, 15], [-49, -24], [38, -44], [-18, 42], [47, 5], [-78, -9], [-88, 0], [-36, 4], [-30, 20], [23, 38], [-60, 1], [16, -11], [23, 15], [-62, -28], [66, 10], [75, -12], [50, 3], [-10, 35], [28, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_ca();".
// 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_6762_2_a_ca(:prec:=2)
    chi := MakeCharacter_6762_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6762_2_a_ca();".
// 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_6762_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<11,R![3, 1]>,<13,R![-2, 0, 1]>,<17,R![-2, 0, 1]>],Snew);
    return Vf;
end function;