
// Make newform 75.3.c.f in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_75_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_75_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_75_3_c_f();".
// 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_75_3_c_f();".
// 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 := [3, -1, 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_75_c();"
function MakeCharacter_75_c()
    N := 75;
    order := 2;
    char_gens := [26, 52];
    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_75_c_Hecke();"
function MakeCharacter_75_c_Hecke(Kf)
    N := 75;
    order := 2;
    char_gens := [26, 52];
    char_values := [[-1, 0], [1, 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 := 3;
    raw_aps := [[1, -2], [3, -1], [0, 0], [0, 0], [-5, 10], [10, 0], [1, -2], [7, 0], [6, -12], [-10, 20], [42, 0], [-40, 0], [5, -10], [-50, 0], [-14, 28], [-14, 28], [-20, 40], [-8, 0], [45, 0], [-10, 20], [-35, 0], [12, 0], [21, -42], [45, -90], [-70, 0], [0, 0], [-70, 0], [21, -42], [-88, 0], [31, -62], [190, 0], [60, -120], [21, -42], [77, 0], [50, -100], [172, 0], [-250, 0], [35, 0], [-54, 108], [-84, 168], [-35, 70], [182, 0], [-70, 140], [-25, 0], [66, -132], [-68, 0], [77, 0], [140, 0], [56, -112], [372, 0], [36, -72], [70, -140], [-413, 0], [-75, 150], [-84, 168], [86, -172], [-140, 280], [22, 0], [-210, 0], [60, -120], [345, 0], [96, -192], [325, 0], [120, -240], [-490, 0], [-64, 128], [-243, 0], [385, 0], [-89, 178], [532, 0], [-84, 168], [-120, 240], [-180, 0], [-110, 0], [-533, 0], [-24, 48], [-30, 60], [-20, 0], [225, -450], [77, 0], [35, -70], [412, 0], [60, -120], [-455, 0], [22, 0], [-159, 318], [25, -50], [-275, 0], [-180, 360], [-240, 0], [-4, 8], [90, -180], [410, 0], [80, -160], [322, 0], [-124, 248], [130, -260], [-85, 170], [1015, 0], [912, 0], [55, 0], [306, -612], [36, -72], [-15, 30], [242, 0], [665, 0], [-189, 378], [-259, 518], [-100, 200], [-343, 0], [-1100, 0], [-290, 0], [-24, 48], [-58, 0], [862, 0], [180, -360], [-1050, 0], [76, -152], [-364, 728], [-245, 490], [-98, 0], [-210, 0], [-24, 48], [51, -102], [-713, 0], [350, -700], [-248, 0], [60, -120], [-10, 0], [770, 0], [802, 0], [406, -812], [322, 0], [400, 0], [-105, 210], [-193, 0], [126, -252], [910, 0], [-334, 668], [380, -760], [-858, 0], [-210, 420], [-1060, 0], [151, -302], [-1038, 0], [-280, 560], [630, 0], [391, -782], [307, 0], [146, -292], [840, 0], [-140, 280], [995, 0], [-144, 288], [-1050, 0], [220, -440], [1342, 0], [140, -280], [-165, 0], [-70, 140], [-84, 168], [-59, 118], [-280, 0], [315, -630], [331, -662], [-504, 1008], [452, 0], [-420, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_75_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_75_3_c_f();".
// 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_75_3_c_f(:prec:=2)
    chi := MakeCharacter_75_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_75_3_c_f();".
// 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_75_3_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_75_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![11, 0, 1]>,<7,R![0, 1]>,<13,R![-10, 1]>],Snew);
    return Vf;
end function;