
// Make newform 2475.2.c.q in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2475_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2475_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_2475_2_c_q();".
// 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_2475_2_c_q();".
// 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 := [8, -20, 25, -4, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [323, 0, -4, 10, 25, 2], [-223, 0, 14, -35, -27, -7], [258, -605, 50, -4, -10, -25], [574, -1331, 372, 38, -26, -65], [-774, 1815, -392, 12, 30, 75]];
    Rf_basisdens := [1, 121, 121, 242, 242, 242];
    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_2475_c();"
function MakeCharacter_2475_c()
    N := 2475;
    order := 2;
    char_gens := [551, 2377, 2026];
    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_2475_c_Hecke();"
function MakeCharacter_2475_c_Hecke(Kf)
    N := 2475;
    order := 2;
    char_gens := [551, 2377, 2026];
    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, 1, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -1], [-1, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 1], [0, 0, 0, 1, 1, 2], [1, 1, -1, 0, 0, 0], [0, 0, 0, 1, 2, -1], [0, 0, 2, 0, 0, 0], [6, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 3], [-1, 0, -1, 0, 0, 0], [0, 0, 0, -4, 2, -3], [0, 0, 0, 3, 0, 1], [0, 0, 0, -2, 4, 4], [-1, 1, -2, 0, 0, 0], [0, -1, -4, 0, 0, 0], [0, 0, 0, 4, 6, 1], [-9, -1, 0, 0, 0, 0], [0, 0, 0, 2, -2, 2], [-1, 2, -1, 0, 0, 0], [0, 0, 0, 4, 6, 0], [2, 0, -4, 0, 0, 0], [0, 0, 0, 11, 6, 2], [1, -2, -3, 0, 0, 0], [0, 0, 0, 4, -4, 0], [0, 0, 0, -2, 0, -4], [0, 1, -2, 0, 0, 0], [0, 0, 0, -6, 0, 0], [0, 0, 0, 3, -1, 4], [-4, 4, 6, 0, 0, 0], [0, 0, 0, -6, 6, 4], [2, -6, -4, 0, 0, 0], [-3, 4, 1, 0, 0, 0], [8, 5, 6, 0, 0, 0], [0, 0, 0, 4, -2, -1], [0, 0, 0, -6, -4, 1], [0, 0, 0, 2, -2, -4], [0, 0, 0, -15, -1, -2], [13, -1, 0, 0, 0, 0], [-1, -6, -8, 0, 0, 0], [-5, -1, 4, 0, 0, 0], [0, 0, 0, -6, 2, 1], [0, 0, 0, 5, 1, 4], [2, 5, 4, 0, 0, 0], [-8, 1, 6, 0, 0, 0], [0, 0, 0, 2, 4, -5], [0, 0, 0, -6, 6, 4], [3, -2, -6, 0, 0, 0], [0, 0, 0, 11, 1, -6], [10, 0, -8, 0, 0, 0], [-6, 3, 4, 0, 0, 0], [4, 2, 0, 0, 0, 0], [0, 0, 0, 6, -4, -10], [0, 0, 0, 2, -4, -4], [-4, 4, -6, 0, 0, 0], [15, -4, -3, 0, 0, 0], [0, 0, 0, 12, 4, 5], [5, 0, -1, 0, 0, 0], [0, 0, 0, -15, 5, 1], [0, 0, 0, -27, 1, 2], [0, 0, 0, 0, -10, -7], [4, 6, -8, 0, 0, 0], [0, 0, 0, 1, 10, 4], [0, 0, 0, -8, -6, -4], [26, 0, 2, 0, 0, 0], [0, 0, 0, 16, -4, 1], [0, 0, 0, 16, -6, 2], [18, 2, 2, 0, 0, 0], [0, 0, 0, -16, -2, 0], [20, 10, 6, 0, 0, 0], [0, 0, 0, 0, -2, -9], [0, 0, 0, 0, -6, -5], [-10, 3, 12, 0, 0, 0], [0, 0, 0, 24, 4, 2], [4, -6, -8, 0, 0, 0], [0, 0, 0, 4, 6, -1], [16, 0, -6, 0, 0, 0], [16, -1, -4, 0, 0, 0], [3, -9, -4, 0, 0, 0], [-9, -13, -2, 0, 0, 0], [-4, 0, -6, 0, 0, 0], [0, 0, 0, 4, 2, -11], [-21, -1, 1, 0, 0, 0], [0, 0, 0, 1, -2, 7], [-14, -12, -6, 0, 0, 0], [0, 0, 0, 2, 6, 2], [0, 4, -2, 0, 0, 0], [0, 0, 0, 6, 8, 6], [0, 0, 0, 16, -4, 2], [0, -2, 10, 0, 0, 0], [0, 0, 0, -20, 4, -7], [-2, -6, 2, 0, 0, 0], [8, -5, -12, 0, 0, 0], [0, 0, 0, 2, -8, 0], [6, 6, 4, 0, 0, 0], [-2, 6, -2, 0, 0, 0], [0, 0, 0, -19, -9, 3], [6, -3, -6, 0, 0, 0], [0, 0, 0, -23, -9, 2], [0, 0, 0, -8, -6, -10], [0, 0, 0, 4, 0, -2], [5, 2, 3, 0, 0, 0], [0, 9, 10, 0, 0, 0], [0, 0, 0, 3, 0, -2], [0, 0, 0, 7, 10, 5], [0, 0, 0, -4, -2, 12], [-3, -9, 2, 0, 0, 0], [6, 5, 0, 0, 0, 0], [0, 0, 0, 6, -4, -2], [0, 0, 0, -2, 4, 4], [0, 0, 0, -4, 6, 14], [-10, -1, 6, 0, 0, 0], [10, -1, 6, 0, 0, 0], [-2, -14, -6, 0, 0, 0], [0, 0, 0, -4, 4, 8], [0, 0, 0, 5, 10, -7], [0, 0, 0, -4, -6, -10], [0, -2, -2, 0, 0, 0], [1, -7, -10, 0, 0, 0], [0, 0, 0, -2, 0, 12], [0, 0, 0, 20, 2, 4], [0, 0, 0, 5, 8, 9], [-6, -12, -10, 0, 0, 0], [21, -8, -9, 0, 0, 0], [7, -2, 6, 0, 0, 0], [0, 14, 0, 0, 0, 0], [0, 0, 0, 6, 10, 11], [0, 0, 0, -22, -18, -6], [-11, -8, 3, 0, 0, 0], [0, 0, 0, 12, 6, 0], [8, -4, 4, 0, 0, 0], [0, 0, 0, 12, -6, -11], [12, 0, 6, 0, 0, 0], [24, 3, -2, 0, 0, 0], [0, 0, 0, 6, -14, -2], [0, 0, 0, -33, 5, 5], [0, 0, 0, -20, 4, -8], [-23, 0, 11, 0, 0, 0], [-17, -1, 1, 0, 0, 0], [4, 10, -10, 0, 0, 0], [0, 0, 0, -12, 2, 3], [0, 0, 0, 36, -4, -6], [14, 4, -8, 0, 0, 0], [8, -6, -4, 0, 0, 0], [0, 0, 0, -36, -8, 1], [0, 0, 0, 1, -19, -6], [12, 2, -14, 0, 0, 0], [0, 0, 0, -24, 8, 12], [0, 0, 0, -6, 4, 5], [-12, 14, -4, 0, 0, 0], [0, 0, 0, 6, 10, 1], [0, 0, 0, -28, -18, -4], [0, 0, 0, -18, 14, 12], [-13, -7, 8, 0, 0, 0], [15, 9, 13, 0, 0, 0], [-2, 4, 6, 0, 0, 0], [0, 0, 0, 34, 2, -1], [7, -4, 13, 0, 0, 0], [0, 0, 0, 1, -4, 1], [0, 0, 0, -11, -9, 0], [0, 0, 0, -26, -8, 6], [-17, -9, -6, 0, 0, 0], [0, 0, 0, -28, 4, -2], [0, 0, 0, -37, -10, -1], [32, -3, -2, 0, 0, 0], [0, 0, 0, 30, 0, -4], [-32, -7, -2, 0, 0, 0], [0, 0, 0, -18, 10, 6], [6, -14, 2, 0, 0, 0], [8, -5, 14, 0, 0, 0], [2, 12, -2, 0, 0, 0], [0, 0, 0, -16, -14, -11], [-8, -1, 20, 0, 0, 0], [6, 2, 6, 0, 0, 0], [-7, 5, -11, 0, 0, 0], [6, 8, -2, 0, 0, 0], [0, 0, 0, 11, -13, -13], [24, 11, -2, 0, 0, 0], [0, 0, 0, -2, -6, -13], [8, 14, 18, 0, 0, 0], [0, 0, 0, -19, 8, 16], [0, 0, 0, -8, 6, -6], [0, 0, 0, 12, -12, -8], [-10, -2, 18, 0, 0, 0], [0, 0, 0, 10, -2, -3], [0, 0, 0, 22, 10, -5], [-52, 1, 6, 0, 0, 0], [-34, -12, -2, 0, 0, 0], [0, 0, 0, 12, -2, -5], [0, 0, 0, 8, -4, 0], [-16, -11, 2, 0, 0, 0], [-8, 0, 20, 0, 0, 0], [0, 0, 0, 18, -2, 12], [0, 0, 0, 16, -12, 4], [-34, -10, -2, 0, 0, 0], [0, 0, 0, 34, 12, 8], [0, 0, 0, -1, -19, 2], [0, 0, 0, -32, -16, -6], [17, -4, -15, 0, 0, 0], [-10, -14, -8, 0, 0, 0], [0, 0, 0, -5, -10, -3], [-30, 0, 0, 0, 0, 0], [-23, -17, 0, 0, 0, 0], [0, 0, 0, 26, 18, 4], [-4, 9, 10, 0, 0, 0], [0, 0, 0, 30, -10, -6], [15, 0, -17, 0, 0, 0], [-26, -1, -2, 0, 0, 0], [0, 0, 0, 14, 2, 7], [-18, 8, -8, 0, 0, 0], [0, 0, 0, 26, 0, -1], [0, 0, 0, -11, -2, 1], [0, 12, 16, 0, 0, 0], [-17, 9, -8, 0, 0, 0], [0, 0, 0, 11, -13, -3], [-19, -12, -5, 0, 0, 0], [0, 0, 0, -35, -8, -7], [0, 0, 0, -10, -10, 10], [-20, -7, 12, 0, 0, 0], [-4, 11, -14, 0, 0, 0], [0, -2, 4, 0, 0, 0], [0, 0, 0, 28, 18, 6], [0, 0, 0, -4, 6, -10], [-24, 5, 24, 0, 0, 0], [0, 0, 0, 6, -10, -2], [19, 5, -2, 0, 0, 0], [0, 0, 0, 15, 15, -5], [10, 12, -18, 0, 0, 0], [0, 0, 0, -1, -6, -4], [-1, 14, 19, 0, 0, 0], [23, -1, 13, 0, 0, 0], [5, 0, -11, 0, 0, 0], [0, 0, 0, 14, 4, 10], [0, 0, 0, 8, 6, 12], [-29, -1, 10, 0, 0, 0], [0, 0, 0, -25, 5, 16], [-29, -3, -8, 0, 0, 0], [-15, -15, -10, 0, 0, 0], [0, 0, 0, -21, 6, 1], [9, 7, 9, 0, 0, 0], [0, 0, 0, 36, -10, 1], [17, -8, -4, 0, 0, 0], [0, 0, 0, -45, 9, 10], [-8, -24, -6, 0, 0, 0], [0, 0, 0, -24, -10, 11], [15, -17, -16, 0, 0, 0], [32, 5, 14, 0, 0, 0], [0, 0, 0, 18, 4, 14], [0, 0, 0, 40, 4, 2], [23, 14, -1, 0, 0, 0], [0, 0, 0, -60, -4, -2], [-28, -17, -10, 0, 0, 0], [0, 0, 0, -48, -10, -6], [36, 16, 8, 0, 0, 0], [19, -4, 18, 0, 0, 0], [0, 0, 0, -30, -4, 2], [0, 0, 0, 2, 6, 4], [0, 0, 0, -14, -12, 12], [0, 0, 0, 9, 21, 11], [0, 0, 0, 6, 24, -12], [22, 10, -10, 0, 0, 0], [0, 0, 0, 18, 16, 7], [0, 0, 0, -4, 2, -8], [0, -6, 14, 0, 0, 0], [34, -6, -28, 0, 0, 0], [16, -14, 14, 0, 0, 0], [0, 0, 0, -31, 5, 0], [0, 0, 0, 49, -1, -2], [-23, 12, -6, 0, 0, 0], [0, 0, 0, 6, 28, 14], [0, 0, 0, -2, -4, -24], [-1, -5, 21, 0, 0, 0], [0, 0, 0, -40, 2, -5], [0, 0, 0, 58, 0, 3], [0, 0, 0, -9, 4, -7], [-20, -3, 12, 0, 0, 0], [-30, -11, 4, 0, 0, 0], [18, 4, -14, 0, 0, 0], [0, 0, 0, 32, -14, -4], [4, 10, 14, 0, 0, 0], [0, 0, 0, -16, 8, 8], [32, -11, -6, 0, 0, 0], [0, 0, 0, -4, -6, -15], [-43, -5, -10, 0, 0, 0], [0, 0, 0, 73, 2, 2], [0, 0, 0, 21, 25, 12], [4, 16, 12, 0, 0, 0], [13, 2, -3, 0, 0, 0], [32, 0, -16, 0, 0, 0], [0, 0, 0, -31, -10, 7], [0, 0, 0, 1, 13, -10], [10, -24, 2, 0, 0, 0], [0, 0, 0, -18, 18, 9], [-8, 0, 6, 0, 0, 0], [0, -7, 2, 0, 0, 0], [0, 0, 0, 24, 8, -4], [-32, 0, -6, 0, 0, 0], [0, 0, 0, -17, -23, -10], [0, 0, 0, -10, 8, -12], [0, 0, 0, 9, 3, 6], [29, -5, -21, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2475_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2475_2_c_q();".
// 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_2475_2_c_q(:prec:=6)
    chi := MakeCharacter_2475_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_2475_2_c_q();".
// 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_2475_2_c_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2475_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![64, 0, 57, 0, 14, 0, 1]>,<7,R![289, 0, 151, 0, 23, 0, 1]>,<29,R![-16, -24, -2, 1]>],Snew);
    return Vf;
end function;