
// Make newform 8575.2.a.a in Magma, downloaded from the LMFDB on 26 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8575_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_8575_2_a_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_8575_2_a_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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    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_8575_a();"
function MakeCharacter_8575_a()
    N := 8575;
    order := 1;
    char_gens := [2402, 3776];
    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_8575_a_Hecke(Kf)
    return MakeCharacter_8575_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 := [[-1, 1, 1], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-1, 5, -3], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-4, 2, -7], [8, -4, 7], [0, 0, 0], [-5, -3, -1], [0, 0, 0], [-2, -5, 2], [0, 0, 0], [-4, -3, 4], [0, 0, 0], [0, 0, 0], [12, -6, 7], [2, -9, -2], [0, 0, 0], [-6, -1, 6], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-4, 2, 7], [4, -11, -4], [-3, -13, 5], [11, 1, 5], [0, 0, 0], [-16, 8, -7], [0, 0, 0], [-9, 17, -13], [13, -9, 11], [0, 0, 0], [6, -13, -6], [0, 0, 0], [0, 0, 0], [-10, 3, 10], [0, 0, 0], [-9, -11, 1], [-16, 8, -21], [-8, 4, 7], [0, 0, 0], [-20, 10, -21], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-8, 15, 8], [-20, 10, -7], [0, 0, 0], [0, 0, 0], [0, 0, 0], [15, -19, 17], [0, 0, 0], [0, 0, 0], [1, 23, -11], [17, -1, 9], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-19, 11, -15], [4, -2, 21], [11, -27, 19], [-4, 0, 0], [0, 0, 0], [0, 0, 0], [8, 0, 0], [0, 0, 0], [24, -12, 7], [-12, 0, 0], [0, 0, 0], [4, 17, -4], [0, 0, 0], [8, 13, -8], [0, 0, 0], [0, 0, 0], [-12, 19, 12], [-10, -11, 10], [0, 0, 0], [0, 0, 0], [20, 0, 0], [16, -9, -16], [-32, 16, -21], [0, 0, 0], [-17, 29, -23], [0, 0, 0], [0, 0, 0], [24, 0, 0], [2, -23, -2], [-23, 3, -13], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-24, 12, -35], [25, -13, 19], [-4, 25, 4], [0, 0, 0], [32, -16, 35], [18, -11, -18], [0, 0, 0], [0, 0, 0], [0, 0, 0], [32, 0, 0], [0, 0, 0], [0, 0, 0], [-13, 37, -25], [-16, -5, 16], [0, 0, 0], [36, -18, 35], [16, -8, 35], [0, 0, 0], [0, 0, 0], [23, -31, 27], [-20, 10, 7], [0, 0, 0], [-16, 23, 16], [0, 0, 0], [-27, 23, -25], [0, 0, 0], [-11, -29, 9], [-20, 13, 20], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-12, 6, -35], [40, 0, 0], [19, -39, 29], [-8, 4, 21], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [27, 5, 11], [0, 0, 0], [-20, -1, 20], [-18, 25, 18], [-44, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-22, 15, 22], [24, -12, -7], [0, 0, 0], [-17, -27, 5], [0, 0, 0], [29, -33, 31], [3, 41, -19], [-48, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-22, 1, 22], [25, -41, 33], [-36, 18, -7], [0, 0, 0], [15, -47, 31], [0, 0, 0], [44, -22, 21], [0, 0, 0], [-48, 24, -35], [0, 0, 0], [-52, 0, 0], [0, 0, 0], [4, -2, 35], [24, -17, -24], [0, 0, 0], [0, 0, 0], [36, -18, 49], [-40, 20, -49], [0, 0, 0], [0, 0, 0], [6, 29, -6], [0, 0, 0], [12, 23, -12], [0, 0, 0], [-44, 22, -49], [0, 0, 0], [-4, -31, 4], [0, 0, 0], [-23, -25, 1], [0, 0, 0], [0, 0, 0], [-2, -33, 2], [22, -29, -22], [0, 0, 0], [15, 37, -11], [0, 0, 0], [16, -8, -21], [24, -12, 49], [0, 0, 0], [0, 0, 0], [-40, 20, -7], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [52, -26, 49], [16, -37, -16], [0, 0, 0], [-5, 53, -29], [0, 0, 0], [-2, 37, 2], [0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 51, -25], [0, 0, 0], [-64, 0, 0], [-2, 0, 0], [-6, 0, 0], [0, 0, 0], [32, -16, -7], [-22, -13, 22], [0, 0, 0], [17, -57, 37], [0, 0, 0], [-29, -23, -3], [0, 0, 0], [4, -2, -35], [-18, 0, 0], [0, 0, 0], [-20, 10, 21], [16, -8, 49], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-22, 0, 0], [-68, 0, 0], [0, 0, 0], [-44, 22, -7], [0, 0, 0], [0, 0, 0], [-37, 45, -41], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [13, 47, -17], [26, -33, -26], [-30, 0, 0], [0, 0, 0], [-41, 37, -39], [0, 0, 0], [0, 0, 0], [33, -53, 43], [-43, 19, -31], [0, 0, 0], [0, 0, 0], [0, 0, 0], [12, -6, 49], [26, 9, -26], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [8, -4, -35], [0, 0, 0], [-36, 18, 7], [38, 0, 0], [0, 0, 0], [-44, 22, -63], [0, 0, 0], [-52, 26, -63], [0, 0, 0], [0, 0, 0], [76, 0, 0], [-40, 20, -63], [19, 45, -13], [0, 0, 0], [0, 0, 0], [-1, 61, -31], [0, 0, 0], [0, 0, 0], [0, 0, 0], [22, -43, -22], [-32, 25, 32], [-8, 4, -49], [0, 0, 0], [0, 0, 0], [46, 0, 0], [0, 0, 0], [-5, -59, 27], [0, 0, 0], [12, -47, -12], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-50, 0, 0], [-12, 6, 35], [-45, 1, -23], [47, -11, 29], [-64, 32, -63], [68, -34, 49], [0, 0, 0], [-34, 13, 34], [-47, 39, -43], [0, 0, 0], [-40, 20, 7], [33, 31, 1], [43, 9, 17], [25, 43, -9], [0, 0, 0], [0, 0, 0], [-10, -39, 10], [0, 0, 0], [16, 33, -16], [0, 0, 0], [-8, -41, 8], [0, 0, 0], [25, -69, 47], [-32, -3, 32], [0, 0, 0], [-6, -43, 6], [0, 0, 0], [52, -26, 7], [-58, 0, 0], [-20, -29, 20], [-4, -45, 4], [0, 0, 0], [0, 0, 0], [41, 19, 11], [0, 0, 0], [17, 55, -19], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-72, 36, -49], [-16, 51, 16], [0, 0, 0], [2, 47, -2], [62, 0, 0], [-26, 47, 26], [0, 0, 0], [34, 1, -34], [0, 0, 0], [0, 0, 0], [0, 0, 0], [24, 25, -24], [31, -71, 51], [0, 0, 0], [36, -29, -36], [0, 0, 0], [0, 0, 0], [0, 0, 0], [9, -73, 41], [-3, 71, -37], [-31, -41, 5], [0, 0, 0], [-44, 22, 7], [92, 0, 0], [0, 0, 0], [-32, 16, 21], [0, 0, 0], [-18, 53, 18], [64, -32, 21], [-38, 17, 38], [0, 0, 0], [23, 53, -15], [-53, 13, -33], [0, 0, 0], [-36, 1, 36], [0, 0, 0], [-76, 38, -49], [0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-64, 32, -77], [96, 0, 0], [-55, 23, -39], [0, 0, 0], [0, 0, 0], [-38, 31, 38], [0, 0, 0], [37, -73, 55], [0, 0, 0], [-48, 24, -77], [0, 0, 0], [-68, 34, -77], [0, 0, 0], [4, -2, -49], [-16, 8, -63], [0, 0, 0], [0, 0, 0], [30, 19, -30], [74, 0, 0], [0, 0, 0], [-47, -17, -15], [6, -55, -6], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-72, 36, -77], [-40, 19, 40], [0, 0, 0], [0, 0, 0], [0, 0, 0], [51, -59, 55], [-100, 0, 0], [-37, -39, 1], [48, -24, -7], [0, 0, 0], [47, -67, 57], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-40, 20, -77], [0, 0, 0], [76, -38, 77], [0, 0, 0], [0, 0, 0], [0, 0, 0], [-22, 57, 22], [-40, 33, 40], [0, 0, 0], [0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8575_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8575_2_a_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_8575_2_a_a(:prec:=3)
    chi := MakeCharacter_8575_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(2999) - 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_8575_2_a_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_8575_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8575_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-13, -4, 3, 1]>,<3,R![0, 1]>],Snew);
    return Vf;
end function;