
// Make newform 525.3.s.f in Magma, downloaded from the LMFDB on 19 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_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_525_3_s_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 := [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, 3, 0, 0], [0, -1, 0, -1], [0, 0, 0, 0], [0, 7, 0, 0], [0, 0, 0, 0], [0, 4, 0, -2], [0, 9, 0, 9], [16, 0, 16, 0], [0, 15, 0, 0], [-6, 0, 0, 0], [26, 0, -13, 0], [0, -70, 0, 0], [-21, 0, 42, 0], [0, 0, 0, -34], [0, 15, 0, -30], [0, -42, 0, 42], [-48, 0, 24, 0], [42, 0, 42, 0], [0, -94, 0, 94], [9, 0, 0, 0], [0, 4, 0, 4], [77, 0, -77, 0], [0, -168, 0, 84], [-33, 0, -33, 0], [0, -114, 0, 57], [192, 0, -96, 0], [0, 75, 0, -150], [0, -192, 0, 0], [0, 0, -128, 0], [0, 0, 0, -183], [0, 0, 0, -10], [-126, 0, -126, 0], [0, -69, 0, 69], [-114, 0, 228, 0], [-156, 0, 156, 0], [0, 0, 86, 0], [0, -98, 0, -98], [0, 188, 0, 0], [0, -72, 0, 36], [0, -78, 0, 156], [0, 0, 270, 0], [24, 0, -48, 0], [-3, 0, 3, 0], [0, 275, 0, -275], [0, 0, 0, 180], [202, 0, -101, 0], [-68, 0, 0, 0], [0, -22, 0, 11], [0, 216, 0, 216], [180, 0, 180, 0], [0, 390, 0, 0], [-357, 0, 0, 0], [-56, 0, 28, 0], [-168, 0, 336, 0], [0, -84, 0, 168], [0, 171, 0, -171], [480, 0, -240, 0], [-137, 0, -137, 0], [0, -130, 0, 130], [-225, 0, 0, 0], [0, -72, 0, -72], [0, -240, 0, 120], [0, 40, 0, -20], [-354, 0, 177, 0], [0, -147, 0, 294], [0, -588, 0, 0], [-70, 0, 70, 0], [0, 0, 0, -611], [0, 90, 0, -90], [136, 0, -272, 0], [0, -87, 0, -87], [-258, 0, 258, 0], [0, -36, 0, -36], [0, 590, 0, 0], [98, 0, 0, 0], [0, -111, 0, 222], [0, 0, 516, 0], [0, -28, 0, 56], [258, 0, -258, 0], [718, 0, -359, 0], [138, 0, -276, 0], [182, 0, 0, 0], [0, 0, -297, 0], [0, 838, 0, -419], [55, 0, 55, 0], [0, -648, 0, 0], [687, 0, 0, 0], [0, 362, 0, 0], [330, 0, -660, 0], [0, 0, 0, -257], [0, 486, 0, -972], [-606, 0, 303, 0], [0, -803, 0, 803], [-678, 0, 0, 0], [-566, 0, 566, 0], [0, 96, 0, -48], [-72, 0, -72, 0], [726, 0, -363, 0], [0, 74, 0, -148], [-44, 0, 44, 0], [0, 0, 0, -376], [0, -78, 0, 78], [0, -336, 0, -336], [237, 0, -237, 0], [0, 0, 580, 0], [0, 236, 0, 236], [0, -1212, 0, 606], [0, -483, 0, 966], [0, 0, -39, 0], [12, 0, -24, 0], [0, 145, 0, -290], [0, -782, 0, 782], [0, 0, 0, -1011], [-116, 0, 58, 0], [395, 0, 0, 0], [0, 0, 309, 0], [0, -568, 0, 284], [0, 684, 0, 684], [0, -510, 0, 0], [-102, 0, 0, 0], [1436, 0, -718, 0], [0, 0, 0, 533], [0, -294, 0, 588], [0, -504, 0, 504], [-208, 0, -208, 0], [-78, 0, 0, 0], [130, 0, -130, 0], [447, 0, 447, 0], [0, 762, 0, -381], [0, -64, 0, 128], [0, 0, -154, 0], [0, 0, 0, -1371], [-526, 0, 526, 0], [0, 0, 0, -1100], [117, 0, 117, 0], [-348, 0, 696, 0], [0, 750, 0, 750], [0, -66, 0, -66], [0, -492, 0, 246], [0, 0, 702, 0], [674, 0, -1348, 0], [-60, 0, 60, 0], [0, -374, 0, 374], [0, 0, 0, -864], [-488, 0, 244, 0], [-525, 0, 1050, 0], [0, -468, 0, 234], [0, -192, 0, -192], [-538, 0, -538, 0], [0, 675, 0, 0], [0, 496, 0, 0], [267, 0, -534, 0], [0, 0, 0, -388], [0, -612, 0, 1224], [0, 1570, 0, -1570], [-555, 0, 0, 0], [-1493, 0, 1493, 0], [528, 0, 528, 0], [0, 776, 0, -388], [84, 0, -42, 0], [0, 1152, 0, 0], [0, 0, 0, 426], [0, 0, 0, 341], [810, 0, 810, 0], [0, -87, 0, 87], [0, 540, 0, 540], [0, 0, 287, 0], [0, 376, 0, 376]];
    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_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_525_3_s_f(: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_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_525_3_s_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![81, 0, -9, 0, 1]>,<11,R![0, 1]>,<13,R![-12, 0, 1]>],Snew);
    return Vf;
end function;