
// Make newform 4025.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4025_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_4025_2_a_f();".
// 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_4025_2_a_f();".
// 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_4025_a();"
function MakeCharacter_4025_a()
    N := 4025;
    order := 1;
    char_gens := [2577, 1151, 3501];
    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_4025_a_Hecke(Kf)
    return MakeCharacter_4025_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], [0], [1], [-4], [2], [-6], [4], [-1], [-2], [4], [10], [10], [-12], [12], [-14], [8], [10], [4], [8], [6], [0], [-12], [-2], [2], [-2], [8], [-12], [-10], [6], [-8], [0], [6], [8], [22], [0], [22], [12], [12], [-6], [12], [10], [-24], [14], [-22], [8], [12], [4], [-4], [-22], [-10], [-8], [6], [-20], [-10], [-16], [30], [28], [-6], [18], [4], [6], [-16], [28], [18], [18], [20], [14], [-36], [-34], [14], [24], [-8], [-14], [12], [-8], [-2], [2], [26], [-14], [36], [22], [-8], [34], [-4], [20], [18], [-34], [-2], [-8], [12], [0], [-8], [12], [28], [-24], [-18], [-42], [-44], [-2], [36], [-38], [44], [18], [20], [-2], [-32], [-26], [0], [-30], [-20], [-22], [38], [-12], [-8], [26], [-4], [-4], [18], [-36], [-14], [-34], [30], [4], [40], [-42], [-18], [20], [8], [-2], [20], [-16], [32], [26], [-30], [-10], [46], [4], [-2], [42], [24], [22], [-8], [-28], [30], [-48], [-22], [14], [40], [0], [50], [-2], [20], [-4], [4], [-16], [-40], [-30], [-38], [-30], [4], [46], [40], [36], [-18], [48], [-16], [-62], [-14], [-54], [20], [14], [48], [-10], [12], [14], [12], [-42], [40], [2], [-8], [52], [26], [38], [48], [10], [34], [52], [10], [-20], [6], [-44], [-20], [-42], [-12], [34], [10], [2], [2], [36], [-50], [-44], [-62], [14], [-52], [58], [-24], [-28], [-6], [40], [14], [30], [-16], [-12], [44], [30], [-40], [10], [24], [-62], [22], [72], [34], [0], [24], [-26], [2], [-24], [-48], [12], [18], [4], [-32], [26], [36], [72], [-10], [66], [-12], [60], [-4], [56], [4], [46], [-10], [4], [12], [-36], [-76], [64], [-6], [70], [48], [6], [2], [12], [-50], [52], [18], [-26], [24], [-36], [-26], [6], [-26], [-12], [30], [-18], [44], [-6], [46], [4], [18], [-32], [-2], [52], [52], [-34], [-70], [-28], [4], [-16], [0], [50], [8], [-24], [6], [-78], [4], [-6], [-34], [-36], [6], [-76], [34], [10], [48], [42], [24], [0], [58], [-78], [-24], [-60], [-12], [46], [-12], [-6], [16], [-38], [12], [78], [10], [-4], [32], [-50], [80], [0], [-54], [18], [-20], [-10], [-58], [-8], [-2], [-42], [60], [-8], [56], [-54], [78], [-30], [-28], [48], [28], [72], [38], [-54], [10], [-40], [-54], [6], [-66], [-48], [-30], [4], [30], [-20], [16], [-42], [-40], [-58], [14], [-8], [-10], [6], [-24], [60], [-18], [-40], [-30], [66], [-16], [-36], [-8], [-26], [34], [16], [-22], [52], [-56], [56], [-42], [-80], [-62], [-8], [72], [34], [-66], [-2], [18], [-66], [-8], [-2], [100], [-4], [96], [-62], [28], [-72], [-86], [2], [-28], [88], [72], [-42], [24], [6], [-76], [-58], [-26], [94], [40], [-22], [-66], [36], [10], [50], [84], [0], [22], [-54], [-76], [60], [22], [30], [-88], [68], [50], [40], [-42], [86], [-8], [-80], [86], [-14], [-60], [-6], [40], [68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4025_2_a_f();".
// 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_4025_2_a_f(:prec:=1)
    chi := MakeCharacter_4025_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_4025_2_a_f();".
// 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_4025_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<3,R![0, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;