
// Make newform 3528.2.a.r in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3528_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_3528_2_a_r();".
// 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_3528_2_a_r();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3528_a();"
function MakeCharacter_3528_a()
    N := 3528;
    order := 1;
    char_gens := [2647, 1765, 785, 1081];
    v := [1, 1, 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_3528_a_Hecke(Kf)
    return MakeCharacter_3528_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+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 := 2;
    raw_aps := [[0], [0], [1], [0], [1], [2], [-3], [5], [3], [6], [-1], [-5], [10], [-4], [-1], [9], [-3], [3], [11], [-16], [7], [-11], [4], [9], [6], [13], [5], [3], [11], [10], [-8], [-17], [-3], [4], [-15], [15], [15], [9], [20], [21], [1], [22], [-17], [-5], [-18], [-9], [12], [24], [-7], [7], [13], [4], [-17], [-24], [13], [-3], [17], [-3], [7], [18], [-17], [-6], [4], [-11], [31], [-27], [-7], [14], [-3], [-6], [5], [15], [19], [19], [-16], [-9], [-19], [-17], [-3], [19], [-20], [10], [41], [-26], [-15], [27], [2], [-17], [6], [-16], [-25], [21], [-13], [0], [-7], [-16], [-7], [-15], [13], [-25], [0], [-11], [-11], [1], [-17], [31], [-12], [-43], [21], [-34], [-7], [-21], [-22], [-5], [0], [-15], [-44], [-43], [5], [36], [-1], [-2], [9], [-39], [-47], [18], [27], [-29], [16], [11], [-41], [-32], [-47], [34], [-27], [14], [-35], [-13], [-42], [25], [-20], [25], [-21], [4], [-37], [24], [-22], [57], [5], [-37], [39], [-18], [28], [55], [-13], [48], [17], [41], [-38], [-39], [-49], [-30], [32], [-57], [37], [33], [59], [31], [-58], [21], [-51], [-14], [-45], [-1], [-17], [-54], [-8], [-7], [0], [-25], [37], [-60], [-58], [-39], [13], [33], [-21], [31], [19], [25], [-1], [44], [-11], [33], [-15], [21], [11], [-33], [-14], [27], [29], [8], [3], [-45], [-20], [-31], [21], [19], [6], [-45], [59], [30], [8], [63], [-15], [-37], [39], [-51], [-9], [2], [-1], [-20], [-7], [-15], [20], [-6], [21], [13], [73], [-25], [-49], [51], [-16], [-19], [28], [37], [11], [73], [36], [28], [-59], [29], [-61], [-61], [30], [51], [56], [41], [-41], [12], [-62], [-15], [-7], [62], [9], [3], [-17], [47], [22], [27], [-29], [-36], [43], [26], [-67], [61], [-50], [58], [-36], [61], [55], [-9], [35], [-71], [54], [-43], [7], [79], [31], [27], [9], [79], [56], [-46], [-3], [55], [47], [-34], [83], [-70], [45], [-63], [-7], [0], [-14], [5], [49], [6], [-41], [-16], [15], [-7], [-25], [84], [-43], [-50], [5], [-38], [-9], [-1], [75], [65], [-35], [67], [-72], [-5], [68], [9], [-2], [10], [-73], [67], [-58], [-20], [-31], [55], [-7], [-55], [3], [-74], [-85], [-19], [40], [-11], [-17], [-28], [66], [-51], [58], [-19], [11], [18], [54], [16], [29], [-76], [-65], [69], [-68], [77], [9], [3], [-10], [-21], [83], [38], [11], [-39], [-3], [56], [18], [61], [-31], [31], [-5], [-21], [-18], [55], [-6], [-3], [5], [-49], [10], [-97], [59], [-15], [0], [-69], [57], [-2], [-51], [42], [-72], [-39], [-60], [9], [-77], [63], [1], [32], [50], [-23], [45], [-7], [-25], [39], [11], [26], [76], [-11], [-5], [-99], [57], [-7], [85], [-83], [-57], [18], [55], [15], [-73], [-3], [-60], [93], [-10], [-35], [71], [103], [-30], [-57], [-15], [-5], [-48], [-100], [-42], [-71], [43], [-30], [-21], [-63]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3528_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_r();".
// 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_3528_2_a_r(:prec:=1)
    chi := MakeCharacter_3528_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_3528_2_a_r();".
// 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_3528_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3528_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<11,R![-1, 1]>,<13,R![-2, 1]>,<23,R![-3, 1]>],Snew);
    return Vf;
end function;