
// Make newform 4225.2.a.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4225_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_4225_2_a_k();".
// 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_4225_2_a_k();".
// 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_4225_a();"
function MakeCharacter_4225_a()
    N := 4225;
    order := 1;
    char_gens := [677, 3551];
    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_4225_a_Hecke(Kf)
    return MakeCharacter_4225_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], [-2], [0], [0], [-2], [0], [0], [6], [6], [-6], [6], [6], [-8], [-6], [-8], [12], [2], [6], [-12], [-2], [6], [0], [-4], [-8], [6], [-6], [6], [-6], [-12], [0], [2], [-12], [2], [4], [-20], [18], [12], [-12], [-16], [12], [-12], [2], [0], [6], [2], [-24], [-12], [24], [4], [-12], [-24], [10], [0], [-12], [0], [-6], [18], [-6], [12], [-8], [-22], [26], [12], [-24], [-8], [2], [-30], [-32], [-6], [12], [-14], [-2], [-18], [-4], [-18], [8], [6], [-18], [-16], [24], [12], [36], [-10], [16], [8], [6], [-16], [30], [-4], [24], [-18], [22], [0], [-12], [6], [-6], [20], [-30], [-42], [-12], [18], [14], [-30], [18], [-12], [-18], [-20], [22], [-24], [-6], [-18], [30], [-34], [18], [-30], [30], [-36], [-6], [36], [-36], [12], [48], [36], [44], [42], [-30], [-12], [0], [26], [-42], [-6], [-16], [32], [20], [40], [0], [38], [12], [-12], [30], [-30], [20], [-42], [4], [6], [46], [-54], [-24], [36], [-8], [-6], [-42], [2], [-42], [-10], [-48], [-24], [-16], [-56], [28], [28], [24], [-48], [12], [-34], [-16], [-16], [-60], [48], [-12], [46], [-12], [24], [-6], [-32], [6], [30], [44], [-42], [18], [-24], [-36], [-4], [-2], [-40], [30], [-12], [-36], [24], [-22], [-48], [-44], [20], [-28], [-18], [-24], [0], [44], [-46], [-54], [-68], [-64], [18], [-26], [2], [60], [-6], [30], [-8], [-12], [16], [-66], [26], [-28], [2], [-24], [14], [18], [32], [-22], [-50], [30], [-64], [-24], [-18], [62], [-24], [-48], [10], [46], [-36], [-52], [-42], [-42], [-42], [56], [0], [10], [-36], [24], [44], [36], [-22], [36], [10], [-24], [72], [70], [-66], [-18], [30], [32], [-64], [-54], [36], [-38], [30], [12], [12], [-42], [46], [-30], [12], [36], [-14], [60], [4], [-40], [12], [36], [18], [36], [-18], [-32], [32], [0], [-52], [12], [-24], [-36], [-78], [18], [6], [-12], [36], [0], [16], [-14], [78], [-6], [54], [-30], [-46], [62], [36], [30], [64], [36], [-60], [12], [-24], [70], [16], [78], [76], [78], [54], [-14], [62], [52], [18], [-68], [-6], [-50], [-16], [30], [70], [86], [-18], [-78], [28], [54], [-90], [72], [-26], [42], [42], [36], [66], [12], [-12], [12], [0], [-92], [18], [-68], [60], [-58], [-72], [18], [78], [72], [4], [8], [-74], [-12], [6], [-60], [-70], [60], [-6], [66], [-20], [46], [-42], [24], [-58], [-62], [-48], [64], [90], [-66], [-18], [14], [-18], [-24], [70], [48], [38], [-60], [-52], [-8], [-30], [-24], [60], [-58], [48], [-6], [-42], [24], [20], [70], [0], [14], [-102], [-32], [54], [76], [36], [78], [48], [-26], [50], [-62], [-14], [40], [-6], [66], [60], [-20], [36], [-48], [24], [10], [4], [-72], [18], [-80], [-60], [-22], [-48], [84], [-42], [44], [24], [-18], [-38], [-10], [98], [54], [-90], [-30], [-8], [-60], [-30], [-46], [-30], [56], [30], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4225_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4225_2_a_k();".
// 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_4225_2_a_k(:prec:=1)
    chi := MakeCharacter_4225_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_4225_2_a_k();".
// 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_4225_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<3,R![2, 1]>,<7,R![0, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;