
// Make newform 4225.2.a.r in Magma, downloaded from the LMFDB on 11 September 2025.
// 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_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_4225_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
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_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 := 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]];
    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, 1], [0, -1], [0, 0], [2, -2], [-2, 1], [0, 0], [2, 2], [-2, -1], [0, 1], [0, 4], [-6, -3], [0, 6], [6, 2], [4, -5], [-2, 2], [6, 6], [-6, -3], [-8, 0], [-2, 0], [-2, 7], [0, -6], [0, 6], [-6, -2], [-6, 0], [-2, 4], [2, 4], [-16, -1], [8, -1], [2, 0], [6, -2], [8, 1], [0, 12], [-6, 8], [-4, -6], [6, -4], [-14, -3], [-18, 0], [2, -12], [-6, -2], [-14, 2], [0, -4], [0, 0], [8, 4], [10, 4], [-6, -12], [4, 0], [-8, -8], [18, 6], [-22, -4], [2, -2], [-2, -16], [2, -1], [6, -6], [0, -14], [-22, 4], [12, 1], [-14, 8], [-14, 9], [10, -2], [-2, 14], [4, -9], [12, -10], [22, -2], [0, -6], [2, -2], [12, 10], [2, -17], [-10, 2], [0, -3], [-10, 6], [12, -2], [18, 7], [-20, 11], [14, 8], [-22, 5], [2, 22], [6, -16], [-8, -14], [-6, -8], [-10, -2], [8, -2], [18, -12], [-22, 13], [-2, 4], [-16, -12], [8, -1], [2, -22], [-18, 0], [-18, -6], [-10, -4], [12, -1], [18, -9], [6, 12], [8, 2], [-26, 11], [-28, 7], [-10, 22], [-12, 4], [24, 3], [10, 2], [8, 15], [-16, 14], [12, 3], [-16, -4], [-20, -6], [-32, 2], [26, 4], [-2, 16], [0, -18], [-22, 16], [-36, -3], [-26, -8], [-6, -12], [-6, -3], [14, 3], [42, -4], [18, 6], [-8, 1], [-30, 12], [32, 10], [10, 28], [2, 6], [6, -10], [22, 8], [-14, -5], [26, 8], [-14, 22], [12, -12], [16, -25], [10, -8], [-18, -9], [-30, -6], [-36, -6], [10, -10], [14, 16], [6, -12], [-8, 10], [14, -6], [6, -12], [0, -32], [10, -13], [6, -32], [-16, -13], [42, -10], [24, -12], [6, 9], [12, 18], [-6, 4], [-8, 14], [-6, 18], [14, -8], [36, 12], [28, 3], [4, -13], [12, 15], [-12, 0], [-8, -6], [-14, 2], [6, -12], [18, 26], [-14, -22], [10, 32], [14, -24], [16, -20], [-48, -6], [-18, -12], [32, -12], [-26, 2], [-2, 28], [6, 18], [18, -11], [10, -6], [-12, 2], [4, -26], [8, -8], [-46, 8], [-6, 15], [-10, 10], [-8, -19], [-26, 8], [-10, -30], [-36, 2], [2, 6], [18, 32], [-6, 8], [-12, -24], [-18, 12], [2, 18], [-2, -30], [18, -23], [14, -26], [18, 10], [-40, -12], [-42, 16], [-36, -15], [30, -12], [-2, 22], [-18, -12], [-36, 2], [-12, -9], [6, 6], [20, 16], [-10, -24], [-24, 0], [-66, -1], [-6, -10], [-6, -21], [32, 7], [-26, -8], [-8, -18], [14, -20], [-24, -8], [-4, -17], [-18, -22], [-18, -35], [10, -18], [-4, 19], [30, -24], [-6, 32], [18, -32], [8, -24], [-2, 25], [42, 8], [46, -2], [-12, -25], [-32, 12], [-22, 4], [4, 2], [-16, -1], [-14, -41], [42, -24], [0, -6], [14, 3], [-2, 28], [24, 15], [38, 16], [-18, -18], [28, -22], [-8, -8], [8, -16], [-18, 18], [-28, 26], [-36, 15], [10, 30], [-54, -4], [16, 8], [6, 30], [-2, -11], [42, -3], [-16, 35], [40, -6], [-18, 24], [-6, -6], [26, 32], [18, -44], [-46, -19], [20, 12], [-26, 4], [26, 4], [8, -18], [-56, -3], [8, 29], [-2, -18], [30, -18], [12, -30], [-52, 18], [-14, 10], [-18, -20], [2, 28], [6, -50], [10, -36], [2, -36], [34, -4], [4, 40], [-14, -42], [-38, 18], [-6, 4], [-38, -18], [-46, 2], [-20, 10], [-12, 23], [-34, -33], [-68, -11], [26, 18], [14, 26], [-12, 2], [22, 16], [46, -4], [-14, 27], [-40, -4], [-72, 0], [56, -1], [30, 16], [22, 5], [-6, 36], [30, 32], [4, 4], [30, 24], [-36, 14], [2, -18], [-26, 12], [14, 20], [-36, 12], [0, -25], [-68, -12], [-16, 22], [28, -17], [34, -12], [-50, -11], [-38, 0], [4, -55], [-6, 30], [20, -20], [24, 33], [-6, 2], [32, 0], [-18, -13], [-18, -37], [0, -42], [-2, 8], [-32, -26], [32, -30], [-32, 8], [-26, 0], [32, 2], [-48, 24], [26, 5], [2, -6], [-8, 17], [-54, -8], [18, 0], [-22, -8], [40, -10], [-6, -36], [-2, 1], [6, -12], [14, -4], [30, 36], [38, -6], [64, 21], [10, 4], [-18, -10], [54, 4], [-16, 22], [34, -20], [8, -40], [50, -8], [2, 30], [-6, 33], [-6, -28], [-6, -21], [18, -12], [54, -28], [-16, 31], [14, -8], [6, 30], [30, -23], [34, -41], [34, -4], [6, -22], [4, 38], [0, 12], [4, -25], [34, -11], [-28, 41], [-46, -26], [46, 20], [-58, 0], [2, -28], [48, 12], [-12, 36], [-30, 24], [-44, -20], [76, -4], [-22, 10], [46, 19], [-48, 30], [-66, 0], [52, -8], [-34, -2], [30, 2], [48, 6], [-30, -6], [40, 14], [38, 15], [6, -4], [-70, -3], [18, -30], [-14, 6], [12, 25], [74, 2], [48, 14], [-46, -23], [-16, 27], [34, -11], [34, -36], [34, -13], [-12, -32], [4, 24], [-26, -28], [-42, -12], [26, -2], [38, 42], [12, 2], [34, 8], [32, -24], [4, 30], [-46, 4], [18, 0], [18, 57], [30, 12], [46, 14], [-24, 9], [-16, -18], [-38, -12], [-42, -16], [18, 7], [-84, -15], [-36, -26], [-48, 35], [-24, -12], [-6, 24], [6, 48], [16, 38], [54, -24], [36, -22], [12, -9], [6, 54], [-30, 3], [16, -38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4225_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4225_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_4225_2_a_r(:prec:=2)
    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_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_4225_2_a_r( : 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, 2, 1]>,<3,R![-2, 0, 1]>,<7,R![-4, -4, 1]>,<11,R![2, 4, 1]>],Snew);
    return Vf;
end function;