
// Make newform 2574.2.a.s in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2574_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_2574_2_a_s();".
// 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_2574_2_a_s();".
// 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_2574_a();"
function MakeCharacter_2574_a()
    N := 2574;
    order := 1;
    char_gens := [1145, 937, 1783];
    v := [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_2574_a_Hecke(Kf)
    return MakeCharacter_2574_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 := [[1], [0], [-2], [-2], [1], [-1], [4], [2], [-4], [-4], [-6], [-2], [2], [-4], [-8], [4], [4], [-14], [2], [0], [-10], [-16], [-12], [10], [14], [-12], [-4], [4], [-14], [-4], [4], [12], [-10], [0], [22], [10], [2], [2], [8], [20], [-12], [-2], [-12], [-6], [-14], [-16], [-12], [26], [-12], [-6], [16], [24], [-10], [0], [12], [0], [24], [-2], [-22], [18], [24], [-2], [-6], [-20], [-6], [22], [-14], [-2], [0], [22], [-18], [-16], [-24], [-10], [-2], [-8], [-12], [-18], [18], [30], [-20], [26], [-8], [30], [24], [-4], [-6], [14], [-22], [-6], [8], [0], [34], [-24], [-38], [0], [-18], [4], [-32], [18], [8], [-18], [36], [-16], [-32], [34], [12], [6], [24], [26], [20], [22], [6], [-10], [6], [-24], [2], [-24], [-8], [-40], [46], [-2], [24], [-44], [-30], [16], [-14], [20], [16], [50], [10], [0], [-12], [-18], [-14], [18], [42], [-6], [-44], [24], [42], [6], [-20], [-20], [14], [-8], [22], [56], [-32], [0], [-10], [-36], [4], [12], [28], [20], [40], [6], [-42], [30], [-36], [0], [22], [16], [42], [-56], [-40], [-42], [-34], [-8], [-36], [-58], [-60], [-18], [4], [16], [-22], [-30], [40], [14], [-62], [-60], [14], [14], [-64], [20], [-18], [-2], [6], [-48], [2], [36], [-20], [18], [-12], [12], [34], [34], [30], [36], [10], [-60], [-14], [26], [36], [-12], [26], [-68], [6], [-60], [54], [-12], [-4], [36], [56], [26], [-40], [20], [8], [34], [-70], [-2], [-30], [-14], [48], [54], [48], [-24], [44], [-36], [-58], [0], [2], [20], [-52], [-72], [-66], [-58], [20], [28], [36], [32], [20], [6], [-2], [20], [-70], [-36], [-14], [28], [66], [2], [-40], [70], [-28], [-44], [26], [-62], [-12], [-62], [44], [8], [-46], [-2], [-78], [-48], [-18], [6], [-62], [52], [-46], [2], [46], [-40], [-2], [70], [-36], [78], [-30], [-56], [-44], [-70], [68], [46], [-22], [48], [-6], [-2], [22], [48], [-52], [8], [-10], [-68], [-38], [-32], [-24], [-44], [48], [-22], [-54], [42], [32], [-64], [76], [54], [44], [50], [40], [-10], [-8], [38], [24], [0], [40], [-26], [-20], [-48], [70], [36], [44], [2], [-72], [46], [86], [-38], [10], [58], [24], [68], [-46], [-72], [-8], [-12], [2], [28], [-34], [-14], [-26], [-32], [70], [28], [-10], [-4], [-26], [28], [-6], [-70], [0], [-4], [6], [-82], [42], [4], [18], [40], [88], [84], [-24], [72], [46], [-36], [72], [76], [-8], [22], [-86], [-54], [-26], [-84], [-64], [32], [-72], [-16], [34], [20], [56], [34], [52], [42], [-6], [54], [78], [-66], [30], [56], [78], [-58], [50], [-40], [-50], [-6], [36], [56], [48], [-86], [-38], [96], [40], [74], [-58], [-48], [62], [34], [10], [40], [-30], [70], [70], [12], [50], [-64], [-24], [-60], [-42], [12], [104], [12], [10], [52], [64], [-90], [-48], [12], [-30], [42], [-96], [22], [-62], [36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2574_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2574_2_a_s();".
// 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_2574_2_a_s(:prec:=1)
    chi := MakeCharacter_2574_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_2574_2_a_s();".
// 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_2574_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2574_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![2, 1]>,<17,R![-4, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;