
// Make newform 112.4.i.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_112_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_112_i_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_112_4_i_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_112_4_i_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_112_i();"
function MakeCharacter_112_i()
    N := 112;
    order := 3;
    char_gens := [15, 85, 17];
    v := [3, 3, 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_112_i_Hecke();"
function MakeCharacter_112_i_Hecke(Kf)
    N := 112;
    order := 3;
    char_gens := [15, 85, 17];
    char_values := [[1, 0], [1, 0], [0, -1]];
    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 := 4;
    raw_aps := [[0, 0], [-5, 5], [0, 9], [21, -14], [-57, 57], [-70, 0], [-51, 51], [0, 5], [0, 69], [114, 0], [23, -23], [0, 253], [-42, 0], [124, 0], [0, 201], [393, -393], [219, -219], [0, 709], [419, -419], [96, 0], [313, -313], [0, 461], [588, 0], [0, 1017], [-1834, 0], [285, -285], [0, -499], [0, -1107], [-923, 923], [1542, 0], [2056, 0], [0, 2049], [141, -141], [-1484, 0], [0, 57], [839, -839], [2833, -2833], [0, -2311], [-1260, 0], [0, -3267], [1287, -1287], [-2674, 0], [0, 4185], [85, -85], [-390, 0], [-2833, 2833], [124, 0], [-56, 0], [-3057, 3057], [0, 961], [0, 2829], [3540, 0], [-5231, 5231], [-5040, 0], [0, 1437], [-2325, 2325], [2385, -2385], [0, -331], [-4871, 4871], [-7026, 0], [-5353, 5353], [4158, 0], [9604, 0], [10131, -10131], [0, -10799], [0, -531], [0, -7015], [8990, 0], [-8709, 8709], [6482, 0], [2133, -2133], [0, 3849], [6491, -6491], [0, -923], [-6344, 0], [0, -5007], [-12291, 12291], [0, -887], [0, -11955], [3421, -3421], [5460, 0], [7730, 0], [-11313, 11313], [4214, 0], [0, 16553], [0, -16395], [-15090, 0], [0, 14785], [2898, 0], [-464, 0], [0, 4233], [2739, -2739], [17051, -17051], [4296, 0], [0, 3401], [-16800, 0], [0, -1839], [-303, 303], [0, -21667], [0, -5039], [2392, 0], [22149, -22149], [-8349, 8349], [0, 15345], [-11593, 11593], [14593, -14593], [15372, 0], [0, 14373], [2547, -2547], [-7042, 0], [0, -22591], [8485, -8485], [-18282, 0], [2291, -2291], [6928, 0], [-24975, 24975], [-9548, 0], [10131, -10131], [0, -16659], [-29556, 0], [-191, 191], [2606, 0], [0, 4209], [24303, -24303], [0, 15041], [24726, 0], [0, 4957], [0, 27669], [13888, 0], [0, -14243], [36959, -36959], [12528, 0], [0, -17767], [-28726, 0], [0, 26469], [5054, 0], [35565, -35565], [-8629, 8629], [20706, 0], [16185, -16185], [11788, 0], [0, 29793], [30323, -30323], [-21156, 0], [5269, -5269], [39816, 0], [14546, 0], [31449, -31449], [0, -24523], [0, -8163], [0, -4367], [-50190, 0], [-12308, 0], [0, 31617], [-13525, 13525], [19248, 0], [0, -8695], [0, -19479], [-12502, 0], [15993, -15993], [0, 44001], [-4002, 0], [-10544, 0], [0, -6183], [-3723, 3723], [-45897, 45897], [6467, -6467], [-23039, 23039]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_112_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_112_4_i_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_112_4_i_a(:prec:=2)
    chi := MakeCharacter_112_i();
    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_112_4_i_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_112_4_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_112_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![25, 5, 1]>],Snew);
    return Vf;
end function;