
// Make newform 7569.2.a.u in Magma, downloaded from the LMFDB on 13 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7569_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_7569_2_a_u();".
// 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_7569_2_a_u();".
// 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, 4, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, 0, 1, 0], [0, -3, 0, 1]];
    Rf_basisdens := [1, 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_7569_a();"
function MakeCharacter_7569_a()
    N := 7569;
    order := 1;
    char_gens := [5888, 1684];
    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_7569_a_Hecke(Kf)
    return MakeCharacter_7569_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, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1, 3, 0, -1], [0, 0, 0, 0], [0, 0, 4, 3], [0, 0, 0, 0], [0, 0, -5, -2], [0, 0, 0, 0], [0, 0, 0, 0], [1, -6, 0, 1], [0, 0, -3, -7], [0, 0, 0, 0], [6, -7, 0, 6], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-4, -5, 0, -4], [0, 0, -7, 2], [0, 0, 0, 0], [0, 0, 8, 9], [0, 0, -7, 3], [0, 0, 0, 0], [0, 0, 0, 0], [11, -8, 0, 11], [0, 0, 0, 0], [-9, -2, 0, -9], [0, 0, 0, 0], [-12, 7, 0, -12], [0, 0, 0, 0], [-7, 13, 0, -7], [0, 0, 0, 0], [0, 0, 0, 0], [13, -10, 0, 13], [0, 0, 0, 0], [0, 0, -9, -14], [0, 0, 12, -1], [3, 11, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 15, 4], [0, 0, 0, 0], [16, -9, 0, 16], [0, 0, 0, 0], [0, 0, 2, -13], [-29, 0, 0, 0], [-6, -11, 0, -6], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-15, 16, 0, -15], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, -19, -7], [0, 0, 0, 0], [0, 0, 6, 19], [0, 0, 0, 0], [0, 0, 1, -17], [0, 0, 0, 0], [16, -19, 0, 16], [0, 0, 0, 0], [21, -10, 0, 21], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, 17, 20], [0, 0, 0, 0], [0, 0, 0, 0], [13, 9, 0, 13], [4, 17, 0, 4], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-23, 11, 0, -23], [0, 0, 0, 0], [0, 0, 23, 15], [0, 0, 0, 0], [0, 0, 21, 20], [0, 0, 0, 0], [0, 0, -11, 13], [0, 0, -5, 18], [0, 0, 0, 0], [0, 0, 0, 0], [-17, -7, 0, -17], [0, 0, 0, 0], [0, 0, 1, -21], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -21, -23], [0, 0, 0, 0], [-18, 25, 0, -18], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -17, 9], [-29, 0, 0, 0], [0, 0, 27, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 26, 5], [19, -27, 0, 19], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-24, -1, 0, -24], [-29, 0, 0, 0], [19, -28, 0, 19], [0, 0, 0, 0], [0, 0, 27, 5], [-1, 0, 0, 0], [0, 0, 0, 0], [-7, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [11, 0, 0, 0], [-13, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, -30], [0, 0, 0, 0], [0, 0, 25, -3], [0, 0, 0, 0], [0, 0, -13, 18], [31, -12, 0, 31], [23, 7, 0, 23], [0, 0, 0, 0], [-10, 31, 0, -10], [-29, 0, 0, 0], [0, 0, 0, 0], [-15, 32, 0, -15], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-31, 6, 0, -31], [0, 0, 0, 0], [0, 0, -33, -19], [0, 0, 0, 0], [-20, 33, 0, -20], [0, 0, 0, 0], [58, 0, 0, 0], [0, 0, 0, 0], [0, 0, -10, -33], [0, 0, 0, 0], [-28, -3, 0, -28], [0, 0, 0, 0], [-21, 34, 0, -21], [0, 0, 0, 0], [0, 0, -26, 7], [0, 0, 0, 0], [35, -18, 0, 35], [0, 0, 0, 0], [35, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -27, -34], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [26, 9, 0, 26], [36, -13, 0, 36], [0, 0, 8, 35], [0, 0, 0, 0], [0, 0, 0, 0], [-11, -25, 0, -11], [0, 0, 0, 0], [58, 0, 0, 0], [22, -37, 0, 22], [0, 0, 0, 0], [41, 0, 0, 0], [0, 0, 0, 0], [0, 0, 34, 31], [-25, -12, 0, -25], [0, 0, 38, 21], [0, 0, 0, 0], [-43, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -37, -9], [-34, 1, 0, -34], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, -23, -39], [0, 0, 0, 0], [0, 0, -25, -39], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-21, -19, 0, -21], [0, 0, 11, -28], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-49, 0, 0, 0], [-29, 0, 0, 0], [0, 0, 13, 40], [0, 0, 0, 0], [0, 0, 0, 0], [33, 5, 0, 33], [0, 0, 0, 0], [0, 0, 0, 0], [15, 26, 0, 15], [0, 0, 7, -32], [0, 0, 0, 0], [41, -14, 0, 41], [0, 0, 0, 0], [0, 0, 0, 0], [9, 31, 0, 9], [0, 0, 23, -19], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -39, -4], [0, 0, -43, -25], [0, 0, 0, 0], [0, 0, 42, 11], [0, 0, 0, 0], [58, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-39, 2, 0, -39], [0, 0, 0, 0], [0, 0, -23, -44], [43, -13, 0, 43], [-29, 0, 0, 0], [0, 0, 0, 0], [-39, 1, 0, -39], [0, 0, 0, 0], [-40, 37, 0, -40], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-45, 19, 0, -45], [0, 0, 9, 43], [0, 0, -17, 28], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 41, 38], [0, 0, 0, 0], [37, 5, 0, 37], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [58, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -44, -35], [0, 0, -6, -43], [0, 0, 0, 0], [0, 0, -24, -47], [-47, 21, 0, -47], [0, 0, 0, 0], [20, 27, 0, 20], [0, 0, -4, 39], [0, 0, 0, 0], [0, 0, -30, 17], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -41, 1], [0, 0, 0, 0], [-5, 44, 0, -5], [0, 0, -14, 33], [-67, 0, 0, 0], [-45, 38, 0, -45], [-31, -17, 0, -31], [0, 0, 9, -37], [0, 0, 0, 0], [-35, 47, 0, -35], [25, 24, 0, 25], [0, 0, 0, 0], [0, 0, 0, 0], [-19, -30, 0, -19], [0, 0, 0, 0], [0, 0, -45, -41], [-29, 0, 0, 0], [0, 0, 0, 0], [49, -33, 0, 49], [0, 0, 0, 0], [0, 0, 50, 23], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-13, 49, 0, -13], [0, 0, 0, 0], [0, 0, 26, 51], [0, 0, 0, 0], [0, 0, 0, 0], [43, 3, 0, 43], [0, 0, 51, 32], [0, 0, 0, 0], [0, 0, 47, 42], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, 48, 7], [0, 0, 0, 0], [-27, 52, 0, -27], [0, 0, 0, 0], [0, 0, 21, 52], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-51, 37, 0, -51], [0, 0, 0, 0], [-5, -43, 0, -5], [0, 0, 0, 0], [0, 0, 0, 0], [77, 0, 0, 0], [0, 0, 0, 0], [0, 0, 9, 50], [0, 0, 32, 53], [0, 0, 0, 0], [0, 0, -43, 6], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, -53, -18], [31, 23, 0, 31], [0, 0, 0, 0], [0, 0, 0, 0], [-49, 4, 0, -49], [0, 0, 0, 0], [0, 0, -38, 15], [0, 0, 0, 0], [0, 0, -19, -54], [0, 0, 0, 0], [28, -55, 0, 28], [0, 0, 0, 0], [0, 0, 24, -31], [54, -17, 0, 54], [58, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -55, -34], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -36, -55], [0, 0, 11, -42], [0, 0, 0, 0], [0, 0, 0, 0], [-46, 51, 0, -46], [-56, 23, 0, -56], [0, 0, 0, 0], [0, 0, -13, 41], [7, 45, 0, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-85, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-57, 23, 0, -57], [56, -17, 0, 56], [0, 0, 0, 0], [0, 0, -51, -2], [-41, -15, 0, -41], [0, 0, 0, 0], [8, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 50], [3, 49, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 41, 57], [0, 0, 0, 0], [-59, 32, 0, -59], [0, 0, 0, 0], [0, 0, 0, 0], [-29, 0, 0, 0], [0, 0, 0, 0], [0, 0, -3, -53], [0, 0, 0, 0], [-54, 5, 0, -54], [-44, -13, 0, -44], [0, 0, -21, 38], [0, 0, 0, 0], [0, 0, 7, 55], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -2, -53], [0, 0, 0, 0], [9, -56, 0, 9], [-28, 0, 0, 0], [0, 0, 0, 0], [55, -49, 0, 55], [0, 0, 0, 0], [0, 0, -37, 23], [0, 0, 0, 0], [0, 0, -17, -59], [0, 0, 0, 0], [0, 0, 0, 0], [-30, 61, 0, -30], [0, 0, 33, 61], [0, 0, 0, 0], [0, 0, -34, 27], [0, 0, 0, 0], [0, 0, 61, 24], [0, 0, 0, 0], [0, 0, 0, 0], [14, 45, 0, 14], [0, 0, -8, 49], [0, 0, 0, 0], [0, 0, 0, 0], [95, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 59, 12], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 11, 59], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 54, 55], [0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7569_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7569_2_a_u();".
// 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_7569_2_a_u(:prec:=4)
    chi := MakeCharacter_7569_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_7569_2_a_u();".
// 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_7569_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7569_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![0, 1]>,<7,R![211, 34, -34, -1, 1]>,<19,R![1711, -94, -94, 1, 1]>],Snew);
    return Vf;
end function;