
// Make newform 1200.3.c.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_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_1200_3_c_i();".
// 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_1200_3_c_i();".
// 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, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 2, -1, -2], [0, 4, 0, 4]];
    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_1200_c();"
function MakeCharacter_1200_c()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_Hecke();"
function MakeCharacter_1200_c_Hecke(Kf)
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [-1, 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, -1, -1, 0], [0, 0, 0, 0], [0, -3, 0, 0], [0, 0, 0, -1], [0, 5, 0, 0], [0, -4, -8, 0], [2, 0, 0, 0], [0, -2, -4, 0], [0, 0, 0, -3], [22, 0, 0, 0], [0, 3, 0, 0], [0, 0, 0, 6], [0, -41, 0, 0], [0, -12, -24, 0], [0, 11, 22, 0], [0, 0, 0, 13], [-86, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 22], [0, 41, 0, 0], [10, 0, 0, 0], [0, -13, -26, 0], [0, 0, 0, 6], [0, 47, 0, 0], [0, 0, 0, 9], [0, 67, 0, 0], [0, -9, -18, 0], [-10, 0, 0, 0], [0, 12, 24, 0], [0, 53, 0, 0], [0, 0, 0, 1], [0, -18, -36, 0], [-78, 0, 0, 0], [0, 0, 0, -29], [-218, 0, 0, 0], [0, 43, 0, 0], [0, 111, 0, 0], [0, 30, 60, 0], [0, 33, 66, 0], [0, 0, 0, 27], [90, 0, 0, 0], [0, 0, 0, 48], [0, 1, 0, 0], [0, -15, -30, 0], [250, 0, 0, 0], [-34, 0, 0, 0], [0, 139, 0, 0], [0, -39, -78, 0], [-58, 0, 0, 0], [0, -70, -140, 0], [0, 0, 0, -4], [-30, 0, 0, 0], [0, 0, 0, 19], [0, 32, 64, 0], [0, -38, -76, 0], [0, 0, 0, -71], [-266, 0, 0, 0], [0, -173, 0, 0], [0, 0, 0, -22], [0, 23, 0, 0], [0, 39, 78, 0], [0, -15, 0, 0], [0, 0, 0, -102], [0, 105, 0, 0], [0, 27, 54, 0], [-434, 0, 0, 0], [0, 255, 0, 0], [0, 27, 54, 0], [-426, 0, 0, 0], [0, -8, -16, 0], [0, 0, 0, -78], [0, -283, 0, 0], [0, 109, 0, 0], [-142, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -97], [0, 155, 0, 0], [0, 0, 0, 60], [270, 0, 0, 0], [0, 0, 0, -9], [-454, 0, 0, 0], [0, 0, 0, -44], [0, 353, 0, 0], [-486, 0, 0, 0], [0, 125, 250, 0], [0, 0, 0, 128], [0, -169, 0, 0], [0, 0, 0, -137], [0, -37, 0, 0], [0, 141, 282, 0], [0, 0, 0, 56], [0, -67, 0, 0], [0, 0, 0, -9], [-30, 0, 0, 0], [0, -42, -84, 0], [0, 0, 0, 21], [0, 0, 0, 14], [0, 247, 0, 0], [234, 0, 0, 0], [0, 145, 0, 0], [0, 31, 62, 0], [0, 43, 86, 0], [0, 0, 0, 110], [-402, 0, 0, 0], [0, -49, 0, 0], [0, -97, -194, 0], [0, -124, -248, 0], [0, 0, 0, -114], [-398, 0, 0, 0], [0, 85, 0, 0], [0, -515, 0, 0], [0, -186, -372, 0], [-14, 0, 0, 0], [-1114, 0, 0, 0], [0, 0, 0, -80], [0, 399, 0, 0], [0, -26, -52, 0], [0, 57, 114, 0], [0, 0, 0, -141], [986, 0, 0, 0], [0, 17, 0, 0], [0, -71, -142, 0], [0, -23, -46, 0], [-578, 0, 0, 0], [0, 0, 0, 219], [1222, 0, 0, 0], [0, 0, 0, -44], [0, -435, 0, 0], [0, -107, 0, 0], [-958, 0, 0, 0], [0, 178, 356, 0], [630, 0, 0, 0], [0, -301, 0, 0], [0, 0, 0, 194], [-770, 0, 0, 0], [0, -33, -66, 0], [0, 257, 0, 0], [0, -125, -250, 0], [0, 0, 0, -134], [1454, 0, 0, 0], [0, 0, 0, -171], [0, 83, 0, 0], [0, -173, -346, 0], [-1258, 0, 0, 0], [0, 0, 0, 234], [0, -371, 0, 0], [0, 42, 84, 0], [-1230, 0, 0, 0], [0, -8, -16, 0], [0, 411, 0, 0], [0, 0, 0, 116], [0, -481, 0, 0], [0, 202, 404, 0], [0, 521, 0, 0], [0, 0, 0, 284], [-614, 0, 0, 0], [0, 0, 0, -8], [0, 231, 0, 0], [0, 0, 0, 63], [0, -123, -246, 0], [0, 270, 540, 0], [0, -35, 0, 0], [0, 0, 0, 111], [0, -180, -360, 0], [0, -114, -228, 0], [854, 0, 0, 0], [0, 259, 0, 0], [958, 0, 0, 0], [0, -37, -74, 0], [0, 0, 0, 285], [426, 0, 0, 0], [0, 0, 0, -314], [0, -583, 0, 0], [-694, 0, 0, 0], [0, 0, 0, -74], [1582, 0, 0, 0], [0, 0, 0, 153], [0, -413, 0, 0], [886, 0, 0, 0], [0, -299, 0, 0], [0, 0, 0, -95], [0, 669, 0, 0], [0, -114, -228, 0], [0, 84, 168, 0], [0, 0, 0, -141], [0, -629, 0, 0], [0, -225, 0, 0], [-1714, 0, 0, 0], [0, 0, 0, 96], [0, 65, 0, 0], [0, 81, 162, 0], [-34, 0, 0, 0], [0, 0, 0, -221], [0, 249, 498, 0], [0, 298, 596, 0], [-1598, 0, 0, 0], [0, 149, 0, 0], [0, 272, 544, 0], [0, -54, -108, 0], [0, 0, 0, 73], [-1034, 0, 0, 0], [0, -781, 0, 0], [-1186, 0, 0, 0], [0, 0, 0, 153], [0, -309, -618, 0], [1706, 0, 0, 0], [0, -369, -738, 0], [0, 0, 0, -446], [494, 0, 0, 0], [0, -849, 0, 0], [0, 0, 0, -3], [0, -365, 0, 0], [0, -197, -394, 0], [0, 0, 0, -94], [1298, 0, 0, 0], [0, 805, 0, 0], [0, 0, 0, -404], [0, 306, 612, 0], [0, -339, -678, 0], [890, 0, 0, 0], [1818, 0, 0, 0], [0, 0, 0, -336], [0, 155, 0, 0], [0, 477, 954, 0], [-794, 0, 0, 0], [0, 150, 300, 0], [0, 0, 0, 360], [0, -195, 0, 0], [0, 0, 0, 263], [0, -507, 0, 0], [1090, 0, 0, 0], [-1898, 0, 0, 0], [0, 0, 0, -242], [0, 951, 0, 0], [0, -212, -424, 0], [766, 0, 0, 0], [0, -397, -794, 0], [0, 0, 0, -283], [0, 0, 0, -178], [0, -69, -138, 0], [-2450, 0, 0, 0], [0, 163, 0, 0], [374, 0, 0, 0], [0, 468, 936, 0], [0, 0, 0, -34], [0, -1003, 0, 0], [0, 0, 0, -327], [-2702, 0, 0, 0], [0, 44, 88, 0], [0, 1131, 0, 0], [0, 0, 0, 240], [0, 502, 1004, 0], [-1138, 0, 0, 0], [0, -279, -558, 0], [0, 0, 0, -93], [1370, 0, 0, 0], [0, 233, 0, 0], [0, 489, 978, 0], [0, -89, 0, 0], [0, -1301, 0, 0], [0, 225, 450, 0], [-2170, 0, 0, 0], [0, -267, 0, 0], [0, 40, 80, 0], [-958, 0, 0, 0], [0, 0, 0, 97], [0, 0, 0, -94], [0, 647, 0, 0], [0, 79, 158, 0], [-2870, 0, 0, 0], [0, 1297, 0, 0], [0, -247, 0, 0], [-1238, 0, 0, 0], [0, 1423, 0, 0], [0, 563, 0, 0], [0, -141, -282, 0], [3094, 0, 0, 0], [-302, 0, 0, 0], [0, 0, 0, 77], [0, -440, -880, 0], [-442, 0, 0, 0], [0, -454, -908, 0], [-1606, 0, 0, 0], [0, 1609, 0, 0], [0, 0, 0, 140], [0, 1105, 0, 0], [0, -243, -486, 0], [2906, 0, 0, 0], [0, 0, 0, 88], [0, 0, 0, -113], [0, -171, -342, 0], [0, -370, -740, 0], [0, 0, 0, 591], [0, 1605, 0, 0], [0, 0, 0, 125], [3670, 0, 0, 0], [0, -597, -1194, 0], [0, 0, 0, 189], [0, -335, 0, 0], [0, 1081, 0, 0], [0, -361, -722, 0], [458, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_i();".
// 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_1200_3_c_i(:prec:=4)
    chi := MakeCharacter_1200_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(1999) - 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_1200_3_c_i();".
// 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_1200_3_c_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![36, 0, 1]>,<11,R![32, 0, 1]>],Snew);
    return Vf;
end function;