
// Make newform 224.3.c.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_c_a();".
// 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_224_3_c_a();".
// 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, 0, 20, 0, 56, 0, 20, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [88, 0, 490, 0, 196, 0, 10, 0], [-48, 97, -77, 77, -21, 21, -1, 1], [-56, -291, 259, -231, 133, -63, 7, -3], [0, -34, 0, 74, 0, 38, 0, 2], [20, 0, 110, 0, 40, 0, 2, 0], [0, -197, 0, -707, 0, -259, 0, -13], [0, -97, 0, -373, 0, -139, 0, -7]];
    Rf_basisdens := [1, 21, 7, 21, 3, 1, 7, 3];
    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_224_c();"
function MakeCharacter_224_c()
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_c_Hecke();"
function MakeCharacter_224_c_Hecke(Kf)
    N := 224;
    order := 2;
    char_gens := [127, 197, 129];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0], [0, -2, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, -3, 0, 0, -2], [0, -1, -1, 1, 0, 0, 1, 0], [0, -4, -1, -1, 0, 0, 0, 0], [-14, 0, 0, 0, 0, -2, 0, 0], [0, 3, 3, -3, 0, 0, 2, 0], [26, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, -2, 0, 0, -6], [0, 4, 5, 5, 0, 0, 0, 0], [0, 3, 3, -3, 0, 0, -2, 0], [-30, 0, 0, 0, 0, 0, 0, 0], [0, -3, -3, 3, 0, 0, -17, 0], [0, 0, 0, 0, 8, 0, 0, -5], [0, 0, -3, -3, 0, 0, 0, 0], [0, 15, 5, 5, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -16], [0, -39, -3, -3, 0, 0, 0, 0], [0, 6, 6, -6, 0, 0, 23, 0], [0, 0, 0, 0, 1, 0, 0, 4], [0, 0, 0, 0, 3, 0, 0, -18], [0, 0, 0, 0, 5, 0, 0, 15], [0, -6, -6, 6, 0, 0, -18, 0], [0, -40, -7, -7, 0, 0, 0, 0], [-78, 0, 0, 0, 0, 4, 0, 0], [-38, 0, 0, 0, 0, -11, 0, 0], [0, 56, -5, -5, 0, 0, 0, 0], [0, 18, 18, -18, 0, 0, -13, 0], [106, 0, 0, 0, 0, -8, 0, 0], [0, 3, 3, -3, 0, 0, 7, 0], [106, 0, 0, 0, 0, -8, 0, 0], [0, -4, 7, 7, 0, 0, 0, 0], [0, 0, 0, 0, 23, 0, 0, 37], [0, -54, -15, -15, 0, 0, 0, 0], [0, -24, -24, 24, 0, 0, 18, 0], [0, 0, 0, 0, -23, 0, 0, 13], [0, 14, -27, -27, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -47], [0, -31, -7, -7, 0, 0, 0, 0], [178, 0, 0, 0, 0, 1, 0, 0], [-86, 0, 0, 0, 0, -2, 0, 0], [0, -2, -2, 2, 0, 0, 22, 0], [0, -12, 21, 21, 0, 0, 0, 0], [0, 8, 8, -8, 0, 0, 48, 0], [0, 21, 21, -21, 0, 0, 65, 0], [0, 0, 0, 0, 17, 0, 0, -17], [-134, 0, 0, 0, 0, -8, 0, 0], [0, 4, 3, 3, 0, 0, 0, 0], [0, 0, 0, 0, 29, 0, 0, -26], [0, -24, -24, 24, 0, 0, -51, 0], [0, 0, 0, 0, 2, 0, 0, -52], [0, -61, 21, 21, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, -31], [0, 18, 18, -18, 0, 0, 72, 0], [-30, 0, 0, 0, 0, 30, 0, 0], [-46, 0, 0, 0, 0, 8, 0, 0], [0, -32, -32, 32, 0, 0, -19, 0], [0, 0, 0, 0, -35, 0, 0, -65], [0, -10, -10, 10, 0, 0, -89, 0], [0, -3, -3, 3, 0, 0, 28, 0], [0, 0, 0, 0, 16, 0, 0, 74], [-70, 0, 0, 0, 0, -10, 0, 0], [0, 10, 5, 5, 0, 0, 0, 0], [250, 0, 0, 0, 0, 5, 0, 0], [0, -76, 21, 21, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, 0, -33], [0, 0, 0, 0, -36, 0, 0, 56], [0, 84, 23, 23, 0, 0, 0, 0], [0, -10, -10, 10, 0, 0, -96, 0], [82, 0, 0, 0, 0, -46, 0, 0], [0, -46, 13, 13, 0, 0, 0, 0], [0, -3, -3, 3, 0, 0, -58, 0], [-294, 0, 0, 0, 0, -18, 0, 0], [0, 0, 0, 0, -32, 0, 0, -73], [-190, 0, 0, 0, 0, 29, 0, 0], [0, 0, 0, 0, -10, 0, 0, 22], [0, 30, 30, -30, 0, 0, -45, 0], [-470, 0, 0, 0, 0, -20, 0, 0], [0, 8, 11, 11, 0, 0, 0, 0], [0, 0, 0, 0, -73, 0, 0, -62], [0, 7, 7, -7, 0, 0, -112, 0], [0, 190, -3, -3, 0, 0, 0, 0], [186, 0, 0, 0, 0, 12, 0, 0], [314, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 2, 0, 0, -37], [0, -91, 25, 25, 0, 0, 0, 0], [0, 27, 27, -27, 0, 0, -23, 0], [0, -45, -45, 45, 0, 0, -150, 0], [0, 100, 23, 23, 0, 0, 0, 0], [0, -128, -51, -51, 0, 0, 0, 0], [0, -54, 57, 57, 0, 0, 0, 0], [0, -45, -45, 45, 0, 0, 104, 0], [0, 0, 0, 0, 38, 0, 0, -33], [0, 0, 0, 0, 98, 0, 0, 42], [0, 63, 63, -63, 0, 0, 151, 0], [-358, 0, 0, 0, 0, 20, 0, 0], [0, 106, -55, -55, 0, 0, 0, 0], [-398, 0, 0, 0, 0, 4, 0, 0], [0, 15, 15, -15, 0, 0, -39, 0], [250, 0, 0, 0, 0, -5, 0, 0], [0, 232, 29, 29, 0, 0, 0, 0], [0, 0, 0, 0, 34, 0, 0, -4], [0, 69, 69, -69, 0, 0, 75, 0], [0, 0, 0, 0, -28, 0, 0, -52], [0, -73, 9, 9, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, 0, -8], [0, -40, -40, 40, 0, 0, -224, 0], [26, 0, 0, 0, 0, -68, 0, 0], [-446, 0, 0, 0, 0, 43, 0, 0], [0, 66, 66, -66, 0, 0, -51, 0], [0, -109, -23, -23, 0, 0, 0, 0], [-286, 0, 0, 0, 0, -7, 0, 0], [0, -17, -17, 17, 0, 0, 61, 0], [0, -18, -18, 18, 0, 0, 134, 0], [-118, 0, 0, 0, 0, 44, 0, 0], [0, 40, -15, -15, 0, 0, 0, 0], [0, 0, 0, 0, 19, 0, 0, 119], [194, 0, 0, 0, 0, -52, 0, 0], [0, 0, 0, 0, -74, 0, 0, -51], [0, -220, 17, 17, 0, 0, 0, 0], [0, -78, -78, 78, 0, 0, 103, 0], [266, 0, 0, 0, 0, -28, 0, 0], [418, 0, 0, 0, 0, 0, 0, 0], [0, 3, 3, -3, 0, 0, 142, 0], [0, 30, 30, -30, 0, 0, -14, 0], [0, 0, 0, 0, -54, 0, 0, 159], [0, 214, -7, -7, 0, 0, 0, 0], [0, -52, 23, 23, 0, 0, 0, 0], [0, -244, -53, -53, 0, 0, 0, 0], [-118, 0, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, 18, 0, 0, -218], [0, 0, 0, 0, -107, 0, 0, 2], [0, 0, 0, 0, -84, 0, 0, -71], [0, -57, -57, 57, 0, 0, -7, 0], [0, 0, 0, 0, 113, 0, 0, 17], [26, 0, 0, 0, 0, 11, 0, 0], [0, 24, 24, -24, 0, 0, 1, 0], [-94, 0, 0, 0, 0, -58, 0, 0], [0, 315, -27, -27, 0, 0, 0, 0], [0, 78, 89, 89, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 147], [0, 42, 42, -42, 0, 0, -242, 0], [0, 0, 0, 0, -21, 0, 0, 111], [0, 0, 0, 0, -100, 0, 0, -70], [0, -4, -4, 4, 0, 0, -131, 0], [0, 137, -103, -103, 0, 0, 0, 0], [-1238, 0, 0, 0, 0, -6, 0, 0], [0, 0, 0, 0, -16, 0, 0, -36], [0, 368, -47, -47, 0, 0, 0, 0], [0, -78, -78, 78, 0, 0, -78, 0], [0, 126, -51, -51, 0, 0, 0, 0], [0, 36, -133, -133, 0, 0, 0, 0], [0, -321, -27, -27, 0, 0, 0, 0], [0, 0, 0, 0, 113, 0, 0, 170], [0, 0, 0, 0, 131, 0, 0, 4], [0, 0, 0, 0, 11, 0, 0, -151], [0, 12, -103, -103, 0, 0, 0, 0], [-1214, 0, 0, 0, 0, -8, 0, 0], [0, 564, -9, -9, 0, 0, 0, 0], [0, -99, -99, 99, 0, 0, 99, 0], [138, 0, 0, 0, 0, 96, 0, 0], [0, -42, -42, 42, 0, 0, -102, 0], [0, 453, 21, 21, 0, 0, 0, 0], [0, 0, 0, 0, -58, 0, 0, -227]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_c_a();".
// 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_224_3_c_a(:prec:=8)
    chi := MakeCharacter_224_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_224_3_c_a();".
// 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_224_3_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![72, 0, 32, 0, 1]>],Snew);
    return Vf;
end function;