
// Make newform 875.2.a.d in Magma, downloaded from the LMFDB on 06 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_875_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_875_2_a_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_875_2_a_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, -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_875_a();"
function MakeCharacter_875_a()
    N := 875;
    order := 1;
    char_gens := [127, 626];
    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;

function MakeCharacter_875_a_Hecke( : Kf := Kf)
    return MakeCharacter_875_a();
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 := 2;
    raw_aps := [[0, 1], [-1, 0], [0, 0], [1, 0], [0, -1], [0, -2], [1, -1], [-2, -1], [2, -1], [-7, 4], [-7, 3], [5, 1], [0, -6], [-3, -1], [-6, -2], [-5, 8], [-8, 1], [-9, -3], [-5, 6], [3, 8], [-8, 9], [-2, -6], [11, -4], [1, -2], [-4, 4], [0, 9], [-5, 8], [-7, 10], [-7, -6], [11, -4], [15, 1], [-1, -4], [-3, -8], [13, -6], [13, -1], [5, 9], [-9, 4], [12, -21], [-1, 8], [-21, 0], [11, -17], [-18, 5], [-9, 12], [-9, 6], [5, -14], [12, -9], [-8, 20], [5, -12], [-2, -10], [-11, -3], [15, -12], [3, 4], [-14, 12], [1, 7], [-6, -12], [0, -12], [1, -2], [-7, -2], [15, -24], [2, -10], [-21, 15], [6, 6], [-14, -1], [22, -10], [-11, 0], [-8, 7], [6, -8], [-9, 19], [6, -6], [-26, -3], [1, 16], [3, 4], [-6, 3], [26, 1], [-12, -6], [30, 3], [0, -15], [-24, -6], [8, -17], [-15, 30], [-3, 26], [-33, 0], [18, -2], [22, -21], [-25, 15], [27, -6], [-9, 18], [17, -3], [-13, 20], [7, 14], [-5, 1], [28, 4], [33, -10], [-11, -14], [2, -9], [-18, 9], [-25, 5], [17, -35], [-5, -22], [0, 9], [11, -21], [-6, 23], [-20, 23], [4, 22], [-11, 6], [11, -26], [-8, 7], [-24, 6], [0, -15], [4, 6], [15, -29], [-4, 6], [29, -22], [21, -12], [6, -18], [21, -23], [-22, -8], [33, -15], [-32, 7], [27, -14], [7, -30], [-9, 6], [-9, 24], [0, 3], [17, 10], [0, -16], [-23, 21], [12, -39], [-21, -12], [32, 9], [43, -6], [0, 13], [9, 21], [46, -6], [39, -19], [18, -6], [40, -17], [-27, 15], [32, 2], [25, -20], [7, 0], [-14, 7], [36, 6], [6, 9], [-2, 9], [30, 0], [13, -33], [38, 10], [-9, 3], [-36, 20], [-9, -26], [-3, 10], [-11, 30], [-21, 38], [12, 2], [-33, 10], [9, -13], [-13, 31], [-33, 37], [-35, 34], [-12, 5], [30, 3], [-12, 0], [37, 5], [-38, -8], [12, 9], [22, -45], [12, 27]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_875_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_875_2_a_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_875_2_a_d(:prec:=2)
    chi := MakeCharacter_875_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_875_2_a_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_875_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_875_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 1]>,<3,R![1, 1]>],Snew);
    return Vf;
end function;