
// Make newform 675.3.i.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_675_i_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_675_3_i_a();".
// 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_675_3_i_a();".
// 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_675_i();"
function MakeCharacter_675_i()
    N := 675;
    order := 6;
    char_gens := [326, 352];
    v := [1, 3];
    // 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_675_i_Hecke();"
function MakeCharacter_675_i_Hecke(Kf)
    N := 675;
    order := 6;
    char_gens := [326, 352];
    char_values := [[1, 0, -1, 0], [-1, 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 := 3;
    raw_aps := [[0, -1, 0, -1], [0, 0, 0, 0], [0, 0, 0, 0], [0, -2, 0, 2], [2, 0, -1, 0], [0, 4, 0, 0], [0, -18, 0, 9], [-11, 0, 0, 0], [0, -16, 0, 32], [52, 0, -26, 0], [-32, 0, 32, 0], [0, 0, 0, 34], [7, 0, 7, 0], [0, -61, 0, 61], [0, -28, 0, -28], [0, 0, 0, 0], [29, 0, 29, 0], [0, 0, -56, 0], [0, -31, 0, 0], [-18, 0, 36, 0], [0, 0, 0, 65], [0, 0, 38, 0], [0, 28, 0, 28], [-72, 0, 144, 0], [0, 115, 0, -115], [-52, 0, 26, 0], [0, 40, 0, 0], [0, 162, 0, -81], [52, 0, 0, 0], [0, -52, 0, 104], [0, 0, 0, 16], [-92, 0, -92, 0], [0, -109, 0, -109], [5, 0, -5, 0], [-88, 0, -88, 0], [0, 0, -20, 0], [0, -40, 0, 0], [0, 0, 0, -106], [0, -110, 0, 220], [0, -134, 0, -134], [-36, 0, 72, 0], [-232, 0, 0, 0], [-268, 0, 134, 0], [0, 265, 0, 0], [0, -144, 0, 72], [-290, 0, 0, 0], [94, 0, -94, 0], [0, -52, 0, 52], [0, -109, 0, -109], [266, 0, -266, 0], [0, 234, 0, -117], [-232, 0, -232, 0], [0, 0, -119, 0], [-225, 0, 450, 0], [0, -101, 0, 202], [0, -26, 0, -26], [108, 0, -216, 0], [-268, 0, 0, 0], [0, -56, 0, 56], [56, 0, -28, 0], [0, -374, 0, 0], [0, 146, 0, -292], [0, 0, 0, -533], [142, 0, 142, 0], [0, 155, 0, -155], [0, -28, 0, -28], [0, 0, -2, 0], [0, 77, 0, 0], [0, -65, 0, 130], [0, 0, 416, 0], [0, 1, 0, 1], [-342, 0, 684, 0], [0, 358, 0, -358], [0, 580, 0, 0], [-83, 0, 0, 0], [0, -322, 0, 644], [-596, 0, 298, 0], [0, 0, 0, -362], [-227, 0, -227, 0], [221, 0, -221, 0], [452, 0, 452, 0], [0, 0, 682, 0], [-162, 0, 324, 0], [0, 0, 0, -295], [0, 0, 812, 0], [0, -53, 0, -53], [369, 0, -738, 0], [0, -65, 0, 65], [920, 0, -460, 0], [0, -734, 0, 0], [0, 234, 0, -117], [700, 0, -350, 0], [0, 0, 0, 106], [133, 0, 133, 0], [-787, 0, 787, 0], [0, 720, 0, -360], [-124, 0, -124, 0], [-117, 0, 234, 0], [0, 0, 0, -250], [650, 0, 0, 0], [0, -623, 0, 623], [0, -612, 0, 306], [0, 65, 0, -130], [754, 0, -377, 0], [-545, 0, 545, 0], [0, 0, 0, 871], [0, -1, 0, -1], [0, 216, 0, -108], [326, 0, 326, 0], [0, 0, -461, 0], [0, -112, 0, 0], [0, 0, 0, 902], [0, 205, 0, -410], [0, 0, -799, 0], [830, 0, 0, 0], [434, 0, -217, 0], [0, 13, 0, 0], [0, -540, 0, 270], [0, 218, 0, -436], [-1136, 0, 568, 0], [382, 0, -382, 0], [0, 578, 0, -578], [0, 404, 0, 404], [0, -1206, 0, 603], [0, 0, -182, 0], [0, 0, 0, 0], [0, 0, -700, 0], [342, 0, -684, 0], [0, 664, 0, -664], [0, 670, 0, 0], [-317, 0, 0, 0], [0, -358, 0, 716], [-1310, 0, 1310, 0], [0, 0, 0, -218], [-380, 0, -380, 0], [1022, 0, -1022, 0], [0, 1368, 0, -684], [0, -130, 0, 0], [0, -182, 0, 364], [81, 0, -162, 0], [299, 0, 0, 0], [-700, 0, 350, 0], [0, 814, 0, 0], [0, 1656, 0, -828], [718, 0, 0, 0], [-920, 0, 460, 0], [0, 1424, 0, -1424], [0, 404, 0, 404], [311, 0, -311, 0], [0, -1188, 0, 594], [0, 104, 0, 0], [36, 0, -72, 0], [0, 0, 0, 119], [0, -686, 0, 1372], [0, -695, 0, 695], [2000, 0, -1000, 0], [-56, 0, 0, 0], [-920, 0, 460, 0], [0, 0, 0, -470], [232, 0, 232, 0], [0, -1, 0, -1], [0, 954, 0, -477], [0, 1202, 0, 0], [-108, 0, 216, 0], [0, 241, 0, -482], [0, -674, 0, -674], [-1420, 0, 0, 0], [0, -524, 0, 524]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_675_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_675_3_i_a();".
// 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_675_3_i_a(:prec:=4)
    chi := MakeCharacter_675_i();
    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_675_3_i_a();".
// 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_675_3_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![9, 0, 3, 0, 1]>],Snew);
    return Vf;
end function;