
// Make newform 525.3.s.d in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_s_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_525_3_s_d();".
// 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_525_3_s_d();".
// 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 := [1, 0, -1, 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_525_s();"
function MakeCharacter_525_s()
    N := 525;
    order := 6;
    char_gens := [176, 127, 451];
    v := [6, 3, 5];
    // 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_525_s_Hecke();"
function MakeCharacter_525_s_Hecke(Kf)
    N := 525;
    order := 6;
    char_gens := [176, 127, 451];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [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 := [[0, 2, 0, 0], [0, 1, 0, 1], [0, 0, 0, 0], [0, -7, 0, 0], [0, 0, -10, 0], [0, -14, 0, 7], [0, -4, 0, -4], [-19, 0, -19, 0], [0, 40, 0, 0], [-16, 0, 0, 0], [6, 0, -3, 0], [0, -5, 0, 0], [14, 0, -28, 0], [0, 0, 0, 19], [0, 30, 0, -60], [0, 32, 0, -32], [-48, 0, 24, 0], [12, 0, 12, 0], [0, 59, 0, -59], [-26, 0, 0, 0], [0, 11, 0, 11], [47, 0, -47, 0], [0, 28, 0, -14], [-68, 0, -68, 0], [0, -56, 0, 28], [-148, 0, 74, 0], [0, 5, 0, -10], [0, 212, 0, 0], [0, 0, 17, 0], [0, 0, 0, -142], [0, 0, 0, -145], [-86, 0, -86, 0], [0, -116, 0, 116], [-49, 0, 98, 0], [124, 0, -124, 0], [0, 0, 46, 0], [0, 108, 0, 108], [0, -58, 0, 0], [0, -308, 0, 154], [0, -72, 0, 144], [0, 0, 10, 0], [189, 0, -378, 0], [2, 0, -2, 0], [0, 235, 0, -235], [0, 0, 0, 100], [232, 0, -116, 0], [2, 0, 0, 0], [0, 392, 0, -196], [0, 94, 0, 94], [-5, 0, -5, 0], [0, -170, 0, 0], [-142, 0, 0, 0], [-176, 0, 88, 0], [-168, 0, 336, 0], [0, 254, 0, -508], [0, -136, 0, 136], [260, 0, -130, 0], [-212, 0, -212, 0], [0, 395, 0, -395], [100, 0, 0, 0], [0, 207, 0, 207], [0, -280, 0, 140], [0, 210, 0, -105], [636, 0, -318, 0], [0, 117, 0, -234], [0, -292, 0, 0], [-5, 0, 5, 0], [0, 0, 0, -439], [0, 220, 0, -220], [196, 0, -392, 0], [0, -178, 0, -178], [292, 0, -292, 0], [0, 311, 0, 311], [0, -205, 0, 0], [523, 0, 0, 0], [0, -44, 0, 88], [0, 0, -74, 0], [0, -187, 0, 374], [128, 0, -128, 0], [-342, 0, 171, 0], [238, 0, -476, 0], [107, 0, 0, 0], [0, 0, -262, 0], [0, 42, 0, -21], [-180, 0, -180, 0], [0, -212, 0, 0], [782, 0, 0, 0], [0, -677, 0, 0], [280, 0, -560, 0], [0, 0, 0, -443], [0, -26, 0, 52], [64, 0, -32, 0], [0, -67, 0, 67], [-68, 0, 0, 0], [509, 0, -509, 0], [0, -756, 0, 378], [-502, 0, -502, 0], [496, 0, -248, 0], [0, -569, 0, 1138], [121, 0, -121, 0], [0, 0, 0, 926], [0, -662, 0, 662], [0, 186, 0, 186], [-758, 0, 758, 0], [0, 0, 865, 0], [0, 619, 0, 619], [0, 392, 0, -196], [0, -142, 0, 284], [0, 0, -284, 0], [-343, 0, 686, 0], [0, -5, 0, 10], [0, -878, 0, 878], [0, 0, 0, -194], [-706, 0, 353, 0], [-250, 0, 0, 0], [0, 0, 1124, 0], [0, 658, 0, -329], [0, 626, 0, 626], [0, -1010, 0, 0], [908, 0, 0, 0], [-834, 0, 417, 0], [0, 0, 0, 1027], [0, 324, 0, -648], [0, -976, 0, 976], [327, 0, 327, 0], [352, 0, 0, 0], [-1150, 0, 1150, 0], [562, 0, 562, 0], [0, 238, 0, -119], [0, 719, 0, -1438], [0, 0, 311, 0], [0, 0, 0, -394], [79, 0, -79, 0], [0, 0, 0, -250], [-548, 0, -548, 0], [497, 0, -994, 0], [0, 130, 0, 130], [0, -144, 0, -144], [0, -1568, 0, 784], [0, 0, -1418, 0], [504, 0, -1008, 0], [-250, 0, 250, 0], [0, -206, 0, 206], [0, 0, 0, 1234], [-398, 0, 199, 0], [-280, 0, 560, 0], [0, -322, 0, 161], [0, -368, 0, -368], [352, 0, 352, 0], [0, 670, 0, 0], [0, 394, 0, 0], [672, 0, -1344, 0], [0, 0, 0, -737], [0, 422, 0, -844], [0, -235, 0, 235], [-740, 0, 0, 0], [1517, 0, -1517, 0], [-642, 0, -642, 0], [0, -966, 0, 483], [-456, 0, 228, 0], [0, 338, 0, 0], [0, 0, 0, 1244], [0, 0, 0, -1741], [740, 0, 740, 0], [0, 262, 0, -262], [0, -640, 0, -640], [0, 0, 67, 0], [0, -571, 0, -571]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_3_s_d();".
// 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_525_3_s_d(:prec:=4)
    chi := MakeCharacter_525_s();
    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_525_3_s_d();".
// 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_525_3_s_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![16, 0, -4, 0, 1]>,<11,R![100, 10, 1]>,<13,R![-147, 0, 1]>],Snew);
    return Vf;
end function;