
// Make newform 3025.2.a.o in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3025_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_3025_2_a_o();".
// 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_3025_2_a_o();".
// 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_3025_a();"
function MakeCharacter_3025_a()
    N := 3025;
    order := 1;
    char_gens := [727, 2301];
    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_3025_a_Hecke(Kf)
    return MakeCharacter_3025_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, 1], [0, 2], [0, 0], [-2, 0], [0, 0], [-4, 2], [4, 2], [0, 0], [0, 2], [-2, 4], [0, 0], [2, 4], [-6, 0], [-6, 0], [0, -2], [-6, -4], [-4, 4], [-2, 8], [-4, -6], [0, 8], [-4, 2], [-4, 0], [-6, 0], [-2, -8], [2, -4], [2, -8], [-4, -2], [2, 4], [2, 4], [-14, -4], [10, -4], [0, 8], [-6, 12], [4, 0], [-6, 4], [12, 0], [14, 0], [-8, -6], [-6, -12], [-8, -10], [4, 4], [10, 8], [-8, 8], [-4, 2], [-8, -2], [16, -4], [16, 0], [8, 2], [14, 8], [-10, 8], [8, -10], [12, 8], [-6, 0], [12, 0], [-2, 8], [6, -12], [6, 8], [4, -8], [4, 2], [-6, -8], [10, 16], [-12, 2], [-22, 4], [16, -8], [-10, 8], [-10, 8], [4, -8], [-12, -6], [-6, 12], [-10, -12], [-10, -8], [-12, 8], [0, 6], [16, 14], [28, 4], [20, -10], [-2, 16], [-2, 8], [6, -8], [-18, 12], [-20, -4], [-6, 0], [0, 8], [2, 4], [16, 0], [24, 2], [6, 16], [-8, 6], [10, -16], [-8, 10], [12, 2], [36, 0], [16, 6], [12, -8], [4, 4], [-6, 16], [-2, -8], [14, 8], [-2, -28], [-6, 0], [-34, 0], [24, -10], [6, -4], [-26, -4], [-12, 24], [10, 12], [-12, 10], [12, -6], [0, -4], [10, -24], [10, 20], [0, -18], [-6, 4], [-20, 4], [0, 0], [30, 0], [-24, 18], [4, -22], [-6, -4], [-12, -24], [22, 16], [-4, -6], [20, 2], [-12, 14], [-28, -8], [-10, 16], [2, 16], [24, -4], [28, -6], [8, -18], [4, -24], [-10, -28], [-16, 0], [-2, -8], [30, 0], [2, 12], [2, -20], [2, -12], [-10, -16], [-6, 12], [-20, -24], [30, 8], [-28, 6], [-6, 20], [18, 0], [32, -4], [-24, 6], [-12, 26], [-4, 20], [12, -2], [-24, -18], [-30, 16], [-40, 2], [10, 20], [36, -6], [24, -24], [32, 0], [-6, 8], [24, -18], [18, -8], [-44, 2], [56, -2], [2, -12], [-20, 8], [-38, 4], [36, -10], [-24, -24], [-16, -18], [26, -4], [18, 12], [28, -8], [26, -8], [-28, -8], [-16, 14], [-8, 12], [-30, -8], [-4, -24], [-26, -16], [18, 24], [-30, -12], [-44, 2], [-4, -16], [14, -24], [-8, -26], [-48, 2], [-30, 0], [-8, 14], [36, 6], [-2, 44], [-24, 16], [18, 12], [-30, 16], [20, 24], [-2, -32], [-34, 16], [-18, 8], [-30, 0], [42, 12], [8, -2], [6, -12], [6, -20], [32, 0], [-38, 12], [-6, -12], [12, -4], [30, -8], [-40, -20], [-42, 12], [18, 20], [-4, -8], [24, 6], [26, 16], [4, -2], [12, 18], [-36, -8], [34, -16], [-58, 0], [-34, 16], [12, -6], [50, 4], [-34, 8], [-20, 32], [58, 8], [-20, -18], [18, -4], [-22, -12], [-30, -20], [8, -44], [-6, 12], [24, 16], [-34, 0], [36, 24], [40, 16], [18, -16], [-28, 14], [14, 32], [-34, 24], [12, 2], [20, -28], [-16, 8], [-24, 18], [-16, 24], [-4, -18], [10, -8], [-32, 22], [20, -32], [-8, 22], [12, -24], [44, 0], [38, 20], [44, 2], [46, 16], [36, 2], [18, 0], [-8, -10], [-44, 16], [34, -24], [14, -40], [14, 32], [-32, -10], [22, 20], [6, 0], [-2, -12], [-20, -14], [22, -16], [28, -4], [-22, -32], [26, -16], [-14, -8], [-12, 26], [14, 48], [-36, 18], [18, 36], [40, 0], [24, -18], [-8, -46], [-20, -14], [-18, -12], [34, 24], [-48, 16], [-10, 20], [8, 24], [30, -24], [10, 24], [14, 0], [16, 32], [10, -36], [-12, 18], [24, 36], [-10, 20], [-2, 32], [-16, 38], [32, -10], [-24, 0], [-20, -6], [42, -4], [0, 24], [42, -4], [60, 16], [2, 36], [32, 14], [-24, 30], [-32, -8], [12, -18], [-20, 0], [-10, -16], [4, 34], [6, -48], [-24, -28], [-8, 30], [34, 8], [-34, -16], [-18, -40], [-28, -10], [-6, -4], [58, -12], [28, 28], [-52, -16], [-62, -12], [-42, -12], [-8, 40], [-14, -20], [6, 48], [0, 42], [-24, 38], [-2, 48], [-4, -12], [-8, 18], [54, 16], [40, -2], [-10, -24], [6, 0], [-12, 24], [6, -12], [32, 32], [-16, 14], [-10, 0], [-16, -2], [22, -16], [-2, 24], [6, 0], [30, -16], [-30, -4], [-48, 0], [-54, 4], [8, -8], [-66, 0], [-8, 6], [-32, -8], [34, 4], [20, 16], [-10, 0], [66, 0], [34, 0], [14, -12], [-36, -30], [-24, -4], [16, 40], [48, 14], [20, 22], [-12, 42], [14, 56], [-8, 38], [16, 40], [-64, -18], [10, -12], [24, -50], [22, -44], [2, -8], [60, 8], [-4, 36], [22, -4], [-54, -20], [32, -8], [6, -24], [12, 44], [4, -32], [-32, -34], [-26, -28], [64, 6], [14, 16], [30, -40], [-14, -36], [-60, -14], [0, -24], [12, -18], [-8, -8], [-42, 0], [-66, -12], [-36, 26], [-82, -8], [-54, -20], [-12, 12], [36, 18], [-48, 32], [-40, -18], [-56, 0], [42, 32], [20, -8], [22, 16], [-70, 12], [-30, 4], [-26, -48], [-42, 0], [-6, -60], [-44, -24], [14, 56], [-18, -40], [-28, -18], [20, -20], [96, 6], [-60, 10], [-40, 2], [44, -8], [20, -22], [-6, 40], [48, 40], [4, -6], [30, -24], [10, 32], [-22, 8], [-84, -6], [-68, -14], [-72, -8], [46, 24], [-30, 12], [-16, 26], [-18, -20], [36, -24], [4, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3025_2_a_o();".
// 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_3025_2_a_o(:prec:=2)
    chi := MakeCharacter_3025_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_3025_2_a_o();".
// 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_3025_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -2, 1]>,<3,R![-8, 0, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;