
// Make newform 675.3.d.k in Magma, downloaded from the LMFDB on 08 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_675_d_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_d_k();".
// 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_d_k();".
// 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 := [4, 0, 64, 0, 16, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 40, 0, 5, 0, 0], [0, 0, 8, 0, 1, 0], [0, -26, 0, -12, 0, -1], [5, 0, 1, 0, 0, 0], [0, 36, 0, 12, 0, 1]];
    Rf_basisdens := [1, 2, 2, 2, 1, 2];
    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_675_d();"
function MakeCharacter_675_d()
    N := 675;
    order := 2;
    char_gens := [326, 352];
    v := [1, 1];
    // 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_d_Hecke();"
function MakeCharacter_675_d_Hecke( : Kf := false)
    N := 675;
    order := 2;
    char_gens := [326, 352];
    char_values := [[-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 := ISA(Type(Kf), Fld), 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]];
    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, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1], [0, -1, 0, 2, 0, -1], [0, -1, 0, -1, 0, -2], [13, 0, -1, 0, -3, 0], [4, 0, -1, 0, 4, 0], [18, 0, -3, 0, 3, 0], [0, 2, 0, -1, 0, 2], [-7, 0, 2, 0, 7, 0], [0, 5, 0, 3, 0, 3], [0, -8, 0, 4, 0, 1], [0, 2, 0, -1, 0, -5], [25, 0, 23, 0, 6, 0], [23, 0, 31, 0, 9, 0], [0, -5, 0, 4, 0, -14], [-13, 0, -11, 0, -16, 0], [0, 0, 0, 5, 0, 4], [0, -11, 0, 7, 0, -11], [0, -16, 0, 8, 0, 1], [-56, 0, 13, 0, -10, 0], [51, 0, 12, 0, -9, 0], [0, 0, 0, -9, 0, 18], [0, -1, 0, 18, 0, -6], [0, 0, 0, 3, 0, 9], [0, 16, 0, -12, 0, 15], [18, 0, 24, 0, 3, 0], [49, 0, 42, 0, -9, 0], [83, 0, -32, 0, 21, 0], [0, 22, 0, -14, 0, 11], [0, 21, 0, -6, 0, 3], [6, 0, 33, 0, 6, 0], [-5, 0, 9, 0, -27, 0], [0, -28, 0, -4, 0, -1], [-103, 0, 25, 0, -25, 0], [0, -16, 0, -6, 0, 12], [0, -50, 0, 6, 0, -9], [-51, 0, -27, 0, 24, 0], [-54, 0, 51, 0, 12, 0], [0, -29, 0, 34, 0, -11], [77, 0, 3, 0, 30, 0], [0, 58, 0, 4, 0, 4], [0, 10, 0, -12, 0, -18], [114, 0, -48, 0, -30, 0], [31, 0, 86, 0, 67, 0], [86, 0, -90, 0, 12, 0], [0, 10, 0, -15, 0, -12], [-46, 0, -74, 0, -69, 0], [52, 0, -59, 0, 5, 0], [105, 0, -63, 0, 39, 0], [0, -23, 0, 1, 0, 31], [-133, 0, -50, 0, 38, 0], [0, 57, 0, 12, 0, -15], [123, 0, 21, 0, -45, 0], [-56, 0, 95, 0, 15, 0], [0, -44, 0, -20, 0, -26], [50, 0, -32, 0, -52, 0], [0, -18, 0, -34, 0, 28], [0, 12, 0, 27, 0, 21], [0, 24, 0, 37, 0, -19], [234, 0, -21, 0, -78, 0], [0, -8, 0, -8, 0, -25], [0, 15, 0, -27, 0, -12], [0, 16, 0, -36, 0, 33], [65, 0, 91, 0, -3, 0], [23, 0, 2, 0, 19, 0], [0, -32, 0, -14, 0, -31], [223, 0, -46, 0, -87, 0], [-164, 0, 14, 0, -44, 0], [216, 0, -111, 0, -42, 0], [0, -39, 0, -9, 0, -12], [0, -6, 0, -34, 0, -44], [0, 20, 0, -46, 0, 61], [-221, 0, -64, 0, 37, 0], [138, 0, 3, 0, 57, 0], [0, 6, 0, -9, 0, -12], [0, 37, 0, 13, 0, -34], [0, -90, 0, 57, 0, -18], [355, 0, 0, 0, 24, 0], [0, 2, 0, 8, 0, 29], [32, 0, -17, 0, -55, 0], [0, 3, 0, -45, 0, 75], [0, 35, 0, -16, 0, 52], [202, 0, 167, 0, 4, 0], [-135, 0, 138, 0, 114, 0], [0, 94, 0, 4, 0, 13], [0, 55, 0, -68, 0, 23], [0, -90, 0, 66, 0, -36], [0, 48, 0, 37, 0, -22], [-82, 0, -32, 0, -57, 0], [0, 0, 0, -48, 0, 9], [0, 44, 0, -39, 0, 57], [0, -38, 0, 4, 0, -83], [94, 0, 8, 0, -110, 0], [-164, 0, -34, 0, -18, 0], [0, -8, 0, 4, 0, -53], [0, -134, 0, 19, 0, -26], [0, 92, 0, -40, 0, -8], [-295, 0, -229, 0, -38, 0], [0, 16, 0, 58, 0, -22], [456, 0, -135, 0, 30, 0], [3, 0, 18, 0, 96, 0], [0, 66, 0, -48, 0, 39], [257, 0, 186, 0, 57, 0], [0, -73, 0, 54, 0, -57], [-348, 0, 198, 0, -138, 0], [160, 0, -52, 0, -126, 0], [0, -16, 0, -82, 0, 11], [-64, 0, 180, 0, 57, 0], [0, 10, 0, -98, 0, 77], [0, 11, 0, -13, 0, -98], [-261, 0, -318, 0, -105, 0], [109, 0, -228, 0, 9, 0], [-145, 0, -113, 0, 29, 0], [0, -36, 0, -33, 0, 72], [0, 24, 0, -56, 0, -7], [-362, 0, 257, 0, -111, 0], [52, 0, -193, 0, 132, 0], [0, 76, 0, 58, 0, -23], [194, 0, -101, 0, 203, 0], [0, -45, 0, 106, 0, -82], [39, 0, -174, 0, -153, 0], [-636, 0, 24, 0, 117, 0], [407, 0, 250, 0, 53, 0], [0, -2, 0, 37, 0, 43], [-125, 0, 119, 0, 118, 0], [0, 15, 0, 33, 0, 96], [0, 164, 0, -6, 0, -48], [0, 1, 0, -3, 0, 39], [-539, 0, 341, 0, -131, 0], [-874, 0, -161, 0, -117, 0], [8, 0, 436, 0, -22, 0], [0, 97, 0, 61, 0, -64], [0, -102, 0, -60, 0, -75], [205, 0, 156, 0, 153, 0], [-219, 0, -405, 0, -177, 0], [0, -182, 0, 34, 0, -70], [551, 0, 235, 0, 159, 0], [0, 122, 0, -67, 0, -13], [-511, 0, -241, 0, 73, 0], [0, -42, 0, -18, 0, -69], [0, 16, 0, -33, 0, 72], [-1157, 0, -214, 0, -60, 0], [-725, 0, 139, 0, 158, 0], [0, -43, 0, 35, 0, 11], [0, 81, 0, 25, 0, 47], [-176, 0, 140, 0, -108, 0], [-26, 0, 341, 0, 82, 0], [16, 0, 326, 0, -108, 0], [0, 147, 0, -19, 0, 109], [0, -136, 0, -64, 0, -28], [0, 28, 0, -93, 0, 120], [-594, 0, 99, 0, 45, 0], [0, 114, 0, -49, 0, 151], [0, 107, 0, 95, 0, 8], [-47, 0, -85, 0, -95, 0], [0, 98, 0, -58, 0, -19], [0, 159, 0, -154, 0, 109], [0, -68, 0, -101, 0, 157], [417, 0, -222, 0, 237, 0], [-1018, 0, 166, 0, -48, 0], [0, -160, 0, 114, 0, -60], [0, 171, 0, 6, 0, 81], [-494, 0, -160, 0, -120, 0], [171, 0, -315, 0, -162, 0], [-880, 0, 234, 0, 42, 0], [0, 63, 0, 17, 0, -101]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_675_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_675_3_d_k();".
// 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_d_k(:prec:=6)
    chi := MakeCharacter_675_d();
    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_d_k();".
// 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_d_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![9, -5, -3, 1]>,<7,R![202500, 0, 10800, 0, 184, 0, 1]>],Snew);
    return Vf;
end function;