
// Make newform 5625.2.a.k in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5625_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_5625_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_5625_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
        poly := [1, 4, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, 0, 1, 0], [0, -3, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_5625_a();"
function MakeCharacter_5625_a()
    N := 5625;
    order := 1;
    char_gens := [4376, 1252];
    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_5625_a_Hecke(Kf)
    return MakeCharacter_5625_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 := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 2], [0, 0, 0, 0], [-1, -3, 0, -1], [0, 0, 0, 0], [-2, 5, 0, -2], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 5, -1], [7, -4, 0, 7], [0, 0, 0, 0], [-6, 7, 0, -6], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -5, -9], [0, 0, 9, 7], [0, 0, 0, 0], [0, 0, -8, -9], [0, 0, 10, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, -3], [0, 0, 0, 0], [-9, -2, 0, -9], [0, 0, 0, 0], [12, -5, 0, 12], [0, 0, 0, 0], [0, 0, 6, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -10, -13], [0, 0, 0, 0], [0, 0, 5, 14], [-25, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [11, -15, 0, 11], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, -15, -2], [-1, 15, 0, -1], [-11, 17, 0, -11], [0, 0, 0, 0], [-17, 5, 0, -17], [0, 0, 0, 0], [0, 0, 0, 0], [-16, 15, 0, -16], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [19, -10, 0, 19], [0, 0, 19, 12], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [-17, -1, 0, -17], [0, 0, 0, 0], [0, 0, 3, 19], [0, 0, 0, 0], [-11, -10, 0, -11], [0, 0, -21, -8], [0, 0, 0, 0], [3, -20, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [-22, 9, 0, -22], [-25, 0, 0, 0], [0, 0, 15, 22], [0, 0, 0, 0], [0, 0, 0, 0], [-12, -11, 0, -12], [0, 0, 0, 0], [0, 0, 15, -8], [0, 0, 0, 0], [0, 0, -20, -21], [0, 0, 0, 0], [0, 0, 13, 24], [-18, -5, 0, -18], [0, 0, 0, 0], [0, 0, 0, 0], [17, -24, 0, 17], [0, 0, 0, 0], [1, -22, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [11, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -17, 9], [17, 0, 0, 0], [13, 14, 0, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -5, 21], [-27, 19, 0, -27], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [23, 0, 0, 0], [0, 0, 26, 23], [0, 0, -28, -19], [0, 0, 0, 0], [27, -5, 0, 27], [14, 15, 0, 14], [0, 0, 0, 0], [0, 0, 18, -11], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -20, 9], [50, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 30, 19], [0, 0, 0, 0], [-31, 0, 0, 0], [0, 0, 0, 0], [0, 0, -31, -13], [50, 0, 0, 0], [-7, 30, 0, -7], [0, 0, 0, 0], [21, 10, 0, 21], [0, 0, 1, -27], [0, 0, 0, 0], [0, 0, -15, -32], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-37, 0, 0, 0], [0, 0, 0, 0], [14, -33, 0, 14], [0, 0, 0, 0], [33, -20, 0, 33], [0, 0, 0, 0], [0, 0, -27, 4], [0, 0, 0, 0], [0, 0, 10, 33], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [21, 13, 0, 21], [0, 0, 0, 0], [33, -26, 0, 33], [0, 0, 0, 0], [0, 0, -35, -18], [0, 0, 0, 0], [-32, 29, 0, -32], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -34, -7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [9, -35, 0, 9], [-13, 36, 0, -13], [0, 0, 35, 8], [0, 0, 0, 0], [0, 0, 0, 0], [47, 0, 0, 0], [0, 0, 0, 0], [-21, 37, 0, -21], [0, 0, 15, 37], [0, 0, 0, 0], [-29, 35, 0, -29], [0, 0, 0, 0], [0, 0, -3, 31], [-49, 0, 0, 0], [0, 0, 21, -17], [0, 0, 0, 0], [0, 0, 7, 36], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-28, -9, 0, -28], [0, 0, 33, 34], [0, 0, -35, -3], [0, 0, 0, 0], [16, 23, 0, 16], [0, 0, 0, 0], [53, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-21, 40, 0, -21], [50, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-39, 10, 0, -39], [0, 0, 4, -33], [0, 0, 40, 27], [0, 0, 0, 0], [0, 0, 0, 0], [-33, -5, 0, -33], [0, 0, 0, 0], [0, 0, 0, 0], [41, -15, 0, 41], [-25, 0, 0, 0], [0, 0, 0, 0], [41, -27, 0, 41], [0, 0, 0, 0], [0, 0, 0, 0], [31, -40, 0, 31], [-23, -19, 0, -23], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [39, -35, 0, 39], [-61, 0, 0, 0], [0, 0, 0, 0], [11, -42, 0, 11], [0, 0, 0, 0], [0, 0, -15, 28], [0, 0, 0, 0], [0, 0, 0, 0], [-37, 39, 0, -37], [0, 0, 0, 0], [0, 0, -23, 21], [-13, -30, 0, -13], [-6, -35, 0, -6], [0, 0, 0, 0], [0, 0, 38, -1], [0, 0, 0, 0], [0, 0, -40, -37], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -45, -26], [-25, 0, 0, 0], [-28, 45, 0, -28], [0, 0, 0, 0], [0, 0, 0, 0], [3, -41, 0, 3], [0, 0, 0, 0], [42, -5, 0, 42], [0, 0, 0, 0], [50, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-13, 45, 0, -13], [0, 0, 0, 0], [0, 0, 0, 0], [-44, 35, 0, -44], [0, 0, -6, 37], [0, 0, 0, 0], [0, 0, 24, -23], [-26, 47, 0, -26], [0, 0, 0, 0], [-27, -20, 0, -27], [0, 0, 43, 39], [0, 0, 0, 0], [47, -30, 0, 47], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -42, -41], [0, 0, 0, 0], [0, 0, 5, 44], [0, 0, 14, 47], [0, 0, -48, -29], [38, -45, 0, 38], [-48, 31, 0, -48], [46, -37, 0, 46], [0, 0, 0, 0], [0, 0, -35, 12], [-73, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 30, 49], [0, 0, 0, 0], [0, 0, -45, -41], [47, -9, 0, 47], [0, 0, 0, 0], [0, 0, -33, 16], [0, 0, 0, 0], [-4, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -13, -49], [0, 0, 0, 0], [77, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [43, -46, 0, 43], [19, 32, 0, 19], [0, 0, 0, 0], [-47, 5, 0, -47], [0, 0, 0, 0], [49, -10, 0, 49], [0, 0, 41, -7], [0, 0, 0, 0], [-79, 0, 0, 0], [0, 0, 0, 0], [0, 0, 52, 21], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 37, 51], [0, 0, 0, 0], [43, 5, 0, 43], [0, 0, 0, 0], [0, 0, 0, 0], [24, -53, 0, 24], [0, 0, 0, 0], [32, 0, 0, 0], [32, 21, 0, 32], [0, 0, 0, 0], [6, 43, 0, 6], [0, 0, 0, 0], [-51, 40, 0, -51], [0, 0, 35, 53], [0, 0, -23, -54], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -45, 4], [0, 0, 0, 0], [0, 0, 15, -38], [0, 0, 0, 0], [54, -35, 0, 54], [0, 0, 0, 0], [-27, 55, 0, -27], [0, 0, 0, 0], [0, 0, 55, 24], [0, 0, -54, -17], [-44, -7, 0, -44], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -55, -34], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -55, -19], [0, 0, 11, 53], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 5, -46], [-23, 56, 0, -23], [0, 0, 0, 0], [-54, 13, 0, -54], [-7, -45, 0, -7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [57, -29, 0, 57], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 34, 57], [-39, -17, 0, -39], [0, 0, 0, 0], [-100, 0, 0, 0], [56, -15, 0, 56], [0, 0, 0, 0], [-91, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-52, 0, 0, 0], [0, 0, -49, -52], [0, 0, 0, 0], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, 59, 32], [0, 0, 0, 0], [0, 0, 0, 0], [-100, 0, 0, 0], [0, 0, 0, 0], [56, 0, 0, 0], [0, 0, 0, 0], [0, 0, -5, -54], [0, 0, 44, -13], [-21, -38, 0, -21], [0, 0, 0, 0], [7, -55, 0, 7], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 51, -2], [0, 0, 0, 0], [0, 0, 47, 56], [0, 0, -15, 43], [0, 0, 0, 0], [0, 0, -55, -49], [0, 0, 0, 0], [0, 0, -60, -23], [0, 0, 0, 0], [-25, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -30, 31], [0, 0, -61, -28], [0, 0, 0, 0], [-34, -27, 0, -34], [0, 0, 0, 0], [0, 0, 37, -24], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 45, -14], [-8, -49, 0, -8], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -29, 33], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-47, 59, 0, -47], [0, 0, 0, 0], [0, 0, 0, 0], [59, -48, 0, 59], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 55, 54], [0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5625_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5625_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_5625_2_a_k(:prec:=4)
    chi := MakeCharacter_5625_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_5625_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_5625_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5625_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![-5, -50, -10, 5, 1]>],Snew);
    return Vf;
end function;