
// Make newform 75.2.g.c in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_75_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_75_g_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_75_2_g_c();".
// 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_75_2_g_c();".
// 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, -11, 96, -409, 877, -179, 236, -22, 34, -2, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1397761481, -35416016020, 198535658181, 45143977692, 49826077334, 15194306845, 7518729011, 2382459144, 688323383, 322545241, 275969449, 78219134], [-16004299508, 716195569825, -2418682513518, 636101258149, -685028180602, 114346528860, -98445654553, 14064780143, -8708425024, 423072977, -2737251577, 295076203], [-6770591262, 22525707095, -92883364032, -350578208249, 3452612852, -101054177500, -1187789257, -14512620193, -101573726, -1432734827, -328887363, -444010163], [-20040639729, 64350556005, 616828423421, -1176010741503, 265393611944, -322352980505, 33133211726, -46835566971, 3118068203, -3917314994, 78219134, -1397761481], [-3685345568, 65280664295, -535050339608, 1546641523369, -268840754862, 424398953870, -31823778603, 61232986233, -2938441244, 5444121987, 240671723, 1818269283], [24741863047, -181257165080, 39335186057, 2963207308274, -235277902802, 837917775445, -19844616263, 122363308068, -1926569149, 11296708427, 1818269283, 3685345568], [51950796787, -557093397120, 3119750034407, -5941261149626, 1312990013398, -1596671748575, 163465627957, -230098529182, 14973917351, -19982886423, 444010163, -6770591262], [81498022997, -557710769585, 2721179313802, -4687952077811, 1100339980428, -1252252255045, 138755057022, -180032467327, 12663738036, -15574701453, 669745048, -5281629612], [-19681913548, 234556298341, -890205653232, 2487936597601, -435400581988, 674811778168, -51790314045, 97434415327, -4686864056, 8742298783, 341056225, 2874546749], [26628527446, -531297349755, 3772098373286, -7678264693623, 1634346533429, -2073953008960, 202386506296, -299493301711, 18549046073, -26000680229, 252041914, -8838097791]];
    Rf_basisdens := [1, 1, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 15154411375, 6061764550, 15154411375];
    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_75_g();"
function MakeCharacter_75_g()
    N := 75;
    order := 5;
    char_gens := [26, 52];
    v := [5, 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_75_g_Hecke();"
function MakeCharacter_75_g_Hecke(Kf)
    N := 75;
    order := 5;
    char_gens := [26, 52];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, -1, 0, 0, 0, 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, 1, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, -1], [0, 0, 1, 1, 3, -1, -1, 0, 0, -2, -1, 1], [0, 0, -1, -2, -1, 0, 1, -3, 0, 0, 2, -1], [0, 0, -1, -2, -1, -2, 1, 0, 2, 1, 1, -1], [4, 0, 0, 0, 2, 2, -1, 2, -4, -1, -2, 2], [-1, 1, 0, 1, -1, -1, -1, 0, 1, 0, 1, 0], [0, 1, 3, 2, 2, 0, 1, -1, 0, 0, -2, 1], [-1, 1, 0, -1, 0, -4, 1, -1, 0, 2, 1, -1], [0, 4, 2, -1, 0, -1, 1, 0, 0, 2, 2, 0], [0, -1, -2, -1, -2, 0, 1, -1, 1, -1, 0, 0], [-3, -1, 0, 0, 0, -3, 0, 0, 3, 1, 0, 1], [4, -1, 0, 0, -2, 4, -2, 0, 1, 1, 0, 1], [-1, -1, 1, 2, 2, 1, -2, -1, 0, -4, -2, 2], [-1, -3, 0, 0, 2, -1, -4, -1, 1, -1, -2, 2], [0, 2, 0, 2, 3, 3, -2, 1, -3, 1, -3, 1], [-2, -5, 0, -1, -1, -2, -1, 0, 4, -2, -1, -2], [-1, -1, -2, -1, -4, 2, 1, 2, 1, 2, 4, -3], [3, -3, 0, -2, -1, 3, -1, 0, 1, -2, -2, -2], [0, 3, 2, 4, 5, 0, -2, 5, 0, -3, -1, 2], [3, 3, -3, -2, -4, -5, 5, -5, -3, 2, 4, -2], [2, 0, 0, -2, -2, 2, -2, 0, -3, 1, -2, 1], [0, -1, -2, -1, -2, 0, -2, 4, -4, -1, 0, 0], [-4, 2, 0, 1, 2, -4, 2, 0, -1, 1, 1, 1], [6, 1, 4, 1, 4, 0, 3, 5, -5, 1, 0, 0], [-1, -2, -3, 0, -2, -2, 0, -1, 0, 0, 1, 1], [4, 0, 1, -1, 1, 2, 1, 4, 0, 2, -1, -3], [3, 0, -5, -1, -2, 8, 1, 8, -3, 1, 2, -1], [0, 1, 0, 2, 2, 0, -2, -9, 0, -1, -1, 1], [1, 2, 3, 0, -1, -7, 0, 1, 0, 0, -1, -1], [0, 0, 2, 0, 0, -8, 0, -8, 0, 0, 0, 0], [-9, -1, 1, -1, 1, 0, 3, 1, -1, -1, 0, 0], [5, -2, 0, -2, -3, 0, 2, 1, -1, 4, 3, -6], [-5, 2, 7, 2, 10, 0, -2, 5, -5, -4, -3, 6], [-2, 6, 6, 0, 3, -5, 0, -2, 0, 0, 0, 0], [0, -4, -1, 4, -2, 2, -5, 0, -2, -5, 1, 2], [2, 4, 5, 6, 8, 3, 0, 3, -2, -4, -8, 2], [1, 2, 0, 0, 3, 1, 3, 0, 2, 4, 0, 4], [0, 2, 3, 0, 2, 8, 3, 3, -8, 2, -2, 0], [2, -1, 1, 1, -2, -1, -1, 2, 0, -2, -2, 0], [-5, 3, -3, 3, -4, 0, 0, 0, 0, 5, 1, -2], [7, -2, 0, -2, 0, 7, 0, 0, 6, -2, -2, -2], [6, 0, 1, 0, 1, 0, -3, 1, -1, 0, 0, 0], [-10, -5, -7, 2, -4, -3, -3, -3, 10, 2, 4, -6], [-1, 0, -5, -6, -6, -7, 3, -7, 1, 3, 6, 0], [-7, -2, 1, 5, 6, 4, -5, 4, 7, -3, -6, 1], [-1, 2, 0, 2, 1, -1, 1, 0, -6, 3, 2, 3], [0, -2, -2, -4, -4, 6, 2, -4, -6, 5, -1, -2], [0, 1, 3, 2, 6, 3, -2, 3, 0, -3, -6, 4], [2, -3, -3, 2, -1, -6, -2, 2, 0, -4, 0, 4], [4, 0, 0, 0, 1, 0, -5, -1, 1, -2, -1, 2], [3, -6, -7, -6, -12, 0, 7, -1, 1, 4, 5, -10], [5, 4, 2, 3, 4, -4, -3, 5, 0, -6, 2, 8], [0, -10, -8, 2, -9, 2, -10, 2, -2, -3, 1, 1], [-1, -3, 0, 0, -7, -1, -7, 0, -5, 0, 0, 0], [10, 0, -1, -1, 4, 4, -2, 4, -10, -2, -4, 5], [0, 4, 1, 2, 5, 9, -1, 7, -9, -3, 1, 1], [0, -7, -9, -10, -12, -6, 1, -9, 6, 2, 8, -5], [-4, -6, 0, -6, -3, 0, 6, 0, 0, 0, 3, -6], [-1, 6, 7, 6, 11, 0, 2, 1, -1, -2, -4, 8], [0, 2, 2, -5, -2, -1, 3, -1, 0, 1, 2, 3], [-7, -2, -12, -7, -16, -10, 7, -7, 0, 14, 10, -4], [0, 5, 7, 6, 8, 11, 1, 4, -11, 3, -9, 3], [0, -6, 0, -8, -10, 3, 8, 1, -3, 3, 5, -4], [-9, 0, -1, -3, -6, 1, 3, -9, 0, 6, 1, -5], [11, 3, 0, 1, -1, 11, -1, 0, 0, -1, 1, -1], [-7, 3, -2, 3, -3, 0, 2, 7, -7, 5, 1, -2], [7, 1, 0, 2, -1, 7, -1, 0, -5, 2, 2, 2], [8, 0, 3, 0, 9, 8, 0, 8, 0, 0, -3, -3], [0, 8, 11, 8, 12, -7, 3, 7, 7, -4, -4, 4], [12, -4, -8, 1, 0, 7, -4, 7, -12, 0, 0, -1], [3, -1, 0, -2, 1, 3, 1, 0, 11, -4, -2, -4], [0, -10, -10, -14, -17, -3, 4, -3, 3, 7, 7, -7], [-13, -4, -9, -4, -12, -10, 2, -10, 13, 6, 12, -8], [0, -10, 0, -2, -1, 0, -1, 0, 4, 1, -2, 1], [3, 6, 5, 6, 10, 0, 1, 9, -9, -4, -5, 10], [11, -1, 0, -1, -2, 2, 1, 11, 0, 2, -1, -3], [0, 16, 7, 12, 22, 4, -5, -2, -4, -7, -5, 6], [-6, 7, 0, 2, -3, -6, -3, 0, 18, -2, 2, -2], [0, -7, -5, -6, -10, -6, 1, -19, 6, -2, 8, -3], [0, 3, 8, 8, 7, 10, 0, 11, -10, -8, 0, 4], [-18, -4, 2, -1, 2, -17, -5, -17, 18, -1, -2, 3], [-17, 2, 0, 2, 1, 0, 2, -17, 17, 0, -1, 2], [-20, -1, 0, -1, -2, 0, 0, -6, 6, 3, 2, -4], [-9, -10, -11, -10, -20, 0, 3, -6, 6, 8, 9, -18], [5, -8, -6, -8, -8, -4, -4, -4, -5, 4, 8, 0], [10, -7, 2, 3, 1, 23, -3, 10, 0, -6, -9, -3], [0, 5, -1, 0, 5, -2, -1, 5, 2, -2, 2, 0], [5, -6, 0, -1, -5, 5, -5, 0, 15, 1, -1, 1], [1, -9, -1, 6, 10, 5, -6, 1, 0, -12, -8, 4], [9, -6, -3, 1, -4, 3, -1, 9, 0, -2, -3, -1], [12, -2, 2, -2, 2, 0, 6, 12, -12, -2, 0, 0], [-9, 10, 0, 2, 4, -9, 4, 0, -2, -1, 2, -1], [7, 3, 3, 3, 6, -15, -3, 7, 0, -6, 0, 6], [22, -3, 0, 1, -4, 22, -4, 0, 0, 0, 1, 0], [0, 2, 4, 3, 8, -11, -2, -11, 0, -4, -8, 5], [-5, 6, 14, 6, 16, -6, -6, -5, 0, -12, -8, 4], [5, -8, 0, -1, -1, 5, -1, 0, -19, 1, -1, 1], [-17, 8, 3, 8, 9, 0, -9, 2, -2, -4, -6, 12], [-3, 1, -6, -6, -3, -9, 6, -3, 0, 12, 7, -5], [0, -1, -2, -8, -5, -3, 6, -13, 3, 7, 1, -4], [-16, -2, 0, 1, -4, -16, -4, 0, -5, 7, 1, 7], [8, -1, 1, -5, -6, -12, 2, -12, -8, 3, 6, -1], [-18, -1, 0, -1, 3, -3, 1, -18, 0, 2, -1, -3], [-17, 7, 7, 7, 7, 0, 0, 5, -5, 7, 0, 0], [3, -2, -10, -2, -10, 0, 1, 0, 0, -2, 0, 0], [-6, 4, 0, 4, 2, 0, 4, -13, 13, 0, -2, 4], [21, 5, -2, 5, 3, 0, -11, 8, -8, -5, -5, 10], [1, 5, 7, 5, 0, -7, -5, 1, 0, -10, -2, 8], [0, -11, -8, -4, -13, 8, -4, 9, -8, -4, 8, -2], [0, -13, -6, -10, -18, -15, 4, -16, 15, 0, 10, -5], [0, -3, -7, 4, -1, -1, -11, -6, 1, 3, -7, 2], [-3, -5, -6, 0, 2, -10, 0, -3, 0, 0, 1, 1], [6, -2, 4, -2, 7, 0, -5, -9, 9, -8, -3, 6], [-2, 0, 0, -5, -4, -2, -4, 0, 9, 1, -5, 1], [11, 5, 0, -5, 2, 11, 2, 0, 1, -3, -5, -3], [-8, 2, 1, -2, 7, -4, 2, -8, 0, 4, 1, -3], [-16, 6, -8, -6, -8, -28, 10, -28, 16, 4, 8, -2], [-7, -1, -3, -2, -2, 13, 0, 13, 7, 1, 2, 0], [23, -4, 7, 6, 8, 21, -8, 21, -23, -4, -8, 2], [0, -1, 2, 6, 2, 8, -4, 25, -8, -7, 1, 3], [17, 2, 3, 3, 8, 19, -3, 17, 0, -6, -1, 5], [-4, -5, -4, -5, -10, 0, 5, 0, 0, 7, 6, -12], [-7, -1, -6, 1, -5, -10, -1, -7, 0, -2, 5, 7], [0, -17, -4, -8, -21, 2, 4, -1, -2, 1, 7, -4], [-16, 3, 13, 12, 14, 1, -4, 1, 16, -7, -14, 2], [0, -1, -6, -12, -7, -1, 6, -30, 1, 6, 6, -6], [-12, -1, -9, 5, -8, -6, 3, -6, 12, 4, 8, -13], [-14, 1, -2, 1, -4, 0, 9, -9, 9, 5, 2, -4], [-4, -7, 3, -7, 0, 0, 9, -9, 9, -1, 3, -6], [4, -2, 4, -2, 2, 0, -12, 3, -3, 2, 2, -4], [-22, 3, 9, 13, 12, 7, -3, 7, 22, -6, -12, -1], [0, 7, 12, 14, 14, -12, -2, 7, 12, -5, -9, 7], [6, 2, 0, -8, -2, -3, 3, -3, -6, 1, 2, 6], [1, -1, -6, -6, -11, -1, 6, 1, 0, 12, 5, -7], [7, 21, 0, 1, 12, 7, 12, 0, -2, 5, 1, 5], [16, 6, 14, 9, 20, 8, -9, 16, 0, -18, -8, 10], [13, -2, -5, -6, -6, 13, 1, 13, -13, 3, 6, 0], [14, -19, 0, -4, -6, 14, -6, 0, -7, -4, -4, -4], [16, -4, 8, 13, 14, 19, -11, 19, -16, -7, -14, 1], [7, 1, -4, -5, -2, -21, 2, -21, -7, 1, 2, 3], [4, 14, 0, 3, 8, 4, 8, 0, 3, -2, 3, -2], [23, 5, 10, -2, -4, 2, 7, 2, -23, 2, 4, -2], [5, 19, 0, 7, 10, 5, 10, 0, 4, -1, 7, -1], [-14, 6, 5, 6, 9, 0, -9, -15, 15, -2, -4, 8], [-14, -5, 3, 7, -3, -2, -7, -14, 0, -14, -8, 6], [3, -5, -7, -4, -17, -8, 4, 3, 0, 8, 2, -6], [2, 7, 1, -5, -6, 5, 10, 5, -2, 3, 6, -1], [0, 6, 12, 12, 12, -6, 0, 22, 6, 0, -12, 6], [0, 12, 13, 0, 12, 6, 13, 6, -6, 7, -7, 0], [-12, -3, -3, -3, 3, -10, 3, -12, 0, 6, 0, -6], [-20, -13, -7, -13, -15, 0, 4, -12, 12, 3, 8, -16], [15, 10, 9, 2, 8, 21, 6, 21, -15, -4, -8, 6], [0, 0, -12, -2, -1, 5, -10, -5, -5, -1, 3, -1], [11, -15, 0, -11, 4, 11, 4, 0, -19, -1, -11, -1], [7, -13, -7, 3, -7, 4, -3, 7, 0, -6, -6, 0], [-27, 7, -6, -11, -20, -12, 11, -27, 0, 22, 13, -9], [2, 4, 0, 2, 9, 2, 9, 0, -6, 1, 2, 1], [0, -16, 0, -3, -1, 0, -1, 0, -14, 1, -3, 1], [0, 11, 12, 6, 14, 12, 6, -7, -12, -5, -1, 3], [-12, 5, 0, 6, -3, -12, -3, 0, 17, -5, 6, -5], [-19, 0, -1, 3, 0, -25, 0, -25, 19, 0, 0, -3], [0, 3, 4, 14, 10, 2, -10, 13, -2, -9, -5, 7], [5, -7, -18, -8, -25, 7, 8, 5, 0, 16, 11, -5], [-15, 22, 0, -1, 11, -15, 11, 0, 15, 9, -1, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_75_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_75_2_g_c();".
// 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_75_2_g_c(:prec:=12)
    chi := MakeCharacter_75_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_75_2_g_c();".
// 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_75_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_75_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -14, 96, -321, 852, -96, 91, -8, 34, -3, 8, 0, 1]>],Snew);
    return Vf;
end function;