
// Make newform 14.4.c.a in Magma, downloaded from the LMFDB on 25 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_14_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_14_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_14_4_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_14_4_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, -1, 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_14_c();"
function MakeCharacter_14_c()
    N := 14;
    order := 3;
    char_gens := [3];
    v := [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_14_c_Hecke();"
function MakeCharacter_14_c_Hecke( : Kf := false)
    N := 14;
    order := 3;
    char_gens := [3];
    char_values := [[-1, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 4;
    raw_aps := [[-2, 2], [0, 5], [9, -9], [-7, -14], [0, 57], [-70, 0], [0, -51], [-5, 5], [-69, 69], [114, 0], [0, -23], [253, -253], [-42, 0], [-124, 0], [-201, 201], [0, 393], [0, -219], [709, -709], [0, -419], [-96, 0], [0, 313], [-461, 461], [-588, 0], [1017, -1017], [-1834, 0], [0, 285], [499, -499], [1107, -1107], [0, -923], [1542, 0], [-2056, 0], [-2049, 2049], [0, 141], [1484, 0], [57, -57], [0, -839], [0, 2833], [2311, -2311], [1260, 0], [-3267, 3267], [0, -1287], [-2674, 0], [-4185, 4185], [0, 85], [-390, 0], [0, 2833], [-124, 0], [56, 0], [0, 3057], [961, -961], [2829, -2829], [-3540, 0], [0, -5231], [5040, 0], [1437, -1437], [0, 2325], [0, 2385], [331, -331], [0, -4871], [-7026, 0], [0, 5353], [4158, 0], [-9604, 0], [0, -10131], [-10799, 10799], [-531, 531], [7015, -7015], [8990, 0], [0, 8709], [6482, 0], [0, 2133], [-3849, 3849], [0, -6491], [-923, 923], [6344, 0], [5007, -5007], [0, -12291], [-887, 887], [-11955, 11955], [0, 3421], [-5460, 0], [7730, 0], [0, 11313], [4214, 0], [-16553, 16553], [16395, -16395], [-15090, 0], [14785, -14785], [2898, 0], [464, 0], [-4233, 4233], [0, -2739], [0, -17051], [-4296, 0], [-3401, 3401], [16800, 0], [-1839, 1839], [0, -303], [21667, -21667], [-5039, 5039], [-2392, 0], [0, 22149], [0, 8349], [15345, -15345], [0, 11593], [0, 14593], [-15372, 0], [14373, -14373], [0, -2547], [-7042, 0], [22591, -22591], [0, 8485], [-18282, 0], [0, -2291], [-6928, 0], [0, -24975], [9548, 0], [0, -10131], [-16659, 16659], [29556, 0], [0, -191], [2606, 0], [4209, -4209], [0, -24303], [-15041, 15041], [24726, 0], [4957, -4957], [-27669, 27669], [-13888, 0], [-14243, 14243], [0, -36959], [-12528, 0], [17767, -17767], [-28726, 0], [26469, -26469], [5054, 0], [0, 35565], [0, 8629], [20706, 0], [0, 16185], [-11788, 0], [29793, -29793], [0, -30323], [21156, 0], [0, 5269], [-39816, 0], [14546, 0], [0, 31449], [24523, -24523], [8163, -8163], [-4367, 4367], [-50190, 0], [12308, 0], [-31617, 31617], [0, 13525], [-19248, 0], [8695, -8695], [-19479, 19479], [-12502, 0], [0, 15993], [-44001, 44001], [-4002, 0], [10544, 0], [6183, -6183], [0, -3723], [0, 45897], [0, -6467], [0, -23039]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_14_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_14_4_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_14_4_c_a(:prec:=2)
    chi := MakeCharacter_14_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_14_4_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_14_4_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_14_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![25, -5, 1]>],Snew);
    return Vf;
end function;