
// Make newform 375.2.g.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_375_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_375_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_375_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_375_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], [-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], [25111873119, -76145058940, -2476340056991, 1740835657738, -826296544974, 451494146575, -112688230641, 64978959816, -10545703663, 4372551549, -2131145819, 1905700356], [-24741863047, 181257165080, -39335186057, -2963207308274, 235277902802, -837917775445, 19844616263, -122363308068, 1926569149, -11296708427, -1818269283, -3685345568], [-51950796787, 557093397120, -3119750034407, 5941261149626, -1312990013398, 1596671748575, -163465627957, 230098529182, -14973917351, 19982886423, -444010163, 6770591262], [13852270880, -355758053867, 1992575852018, -3198526129079, 790744249492, -852450509356, 100643733429, -122610276713, 9161303434, -10484886687, 648267081, -3594254357], [-164302304099, 1590614018810, -9866805339769, 18791172393242, -4156383014316, 5055738779435, -518817498679, 728953008894, -47451650867, 63167876941, -1399563761, 21468863544], [45677018676, -1191999052005, 6669444982566, -24698086195263, 3779688498924, -6776249219160, 433510281051, -981450539091, 39572030488, -88476638949, -6153218841, -29133222771]];
    Rf_basisdens := [1, 1, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 30308822750, 6061764550, 30308822750, 30308822750];
    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_375_g();"
function MakeCharacter_375_g()
    N := 375;
    order := 5;
    char_gens := [251, 127];
    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_375_g_Hecke();"
function MakeCharacter_375_g_Hecke(Kf)
    N := 375;
    order := 5;
    char_gens := [251, 127];
    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, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0, -1, -1, 0, 0], [1, 0, 0, -1, 1, 0, 1, -1, 0, 0, -1, -1], [-1, -1, 0, -1, -1, 0, 0, -2, -2, -1, -1, 1], [1, 1, 0, 1, -2, 0, 1, -1, -1, 1, 0, -1], [-3, 0, -1, 0, -1, 0, 0, 1, -3, -1, 0, -1], [-1, 0, 0, -1, 0, -1, -1, 0, -1, -1, 0, 0], [-1, 0, -2, 0, -1, 2, 0, 2, -2, -1, -1, 1], [0, -1, 1, -2, 4, 0, -1, 0, 0, 0, 1, 1], [1, 3, -3, 2, 0, 0, 0, -1, 0, 0, -1, -1], [0, 0, -2, 2, 0, -1, 0, -1, 0, -1, -1, 0], [3, 2, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0], [-4, 2, 0, 2, -4, 2, 0, 0, 2, 0, 1, 0], [-3, -1, 1, -2, 1, 0, -2, 3, 0, 0, 2, 2], [-2, -3, 1, 0, -2, -3, 0, 2, -2, 1, 0, 1], [1, -1, -1, -1, -1, 1, -1, 0, 0, 1, 2, -1], [-2, -2, 0, -1, -3, -1, 1, 0, -2, 1, 2, 0], [-1, 1, 0, 0, -4, 1, 0, 4, -1, 1, 0, 2], [3, 1, 0, -1, 1, -1, 2, 0, 1, 2, 2, 0], [2, -1, 0, -1, -1, 0, 3, 3, 1, 2, -1, -2], [5, 3, 1, 0, -3, 3, 0, 3, 5, 0, 0, -2], [-2, -1, 0, 2, 0, 2, -2, 0, -2, -2, 1, 0], [0, 0, 2, -2, 0, -2, 0, -4, 5, 1, 1, 0], [4, 0, 0, -2, 3, -2, 1, 0, 0, 1, 1, 0], [-6, 0, 4, -4, 0, -3, 0, 5, -4, 1, 1, 0], [-1, -3, 3, -2, -1, 0, -1, 1, 0, 0, 0, 0], [-5, -2, 2, -3, 0, 0, -3, 5, 0, 0, 1, 1], [-4, 0, -4, 0, -9, 0, 0, 9, -4, 0, 0, 1], [-1, 0, 1, 0, 0, -1, -1, 10, -1, -1, 0, 1], [-1, -3, 3, 1, 8, 0, -1, 1, 0, 0, 0, 0], [0, 0, -2, 0, -8, 0, 0, 8, 0, 0, 0, 0], [-9, 0, -1, 1, 0, 3, 0, -1, 2, 1, 1, 0], [8, 0, -3, 3, 0, -1, 3, -4, 3, -1, -1, -3], [8, 0, 4, -4, 0, -1, 3, 2, -3, -1, -1, -3], [2, -6, 6, -3, 5, 0, 0, -2, 0, 0, 0, 0], [2, 6, -3, 6, -5, 3, 5, -2, -3, 2, -3, -2], [2, -4, 1, 0, 1, -4, 0, -1, 2, 2, 0, -4], [1, -2, 0, 3, 1, 3, 0, 0, -6, 0, -4, 0], [0, 2, -3, 2, 6, 3, 2, -3, 6, 0, -2, 0], [1, 0, 0, -4, -2, 0, 0, -1, 0, 0, 1, 1], [4, 0, -2, 2, 0, 1, -1, 1, 3, 4, 4, 1], [-7, -2, 0, 0, -5, 0, -2, 0, 4, -2, -2, 0], [6, 0, -1, 1, 0, -3, 0, -1, 1, 0, 0, 0], [-8, -5, 5, 0, -1, -5, 0, 1, -8, -4, 0, -2], [-2, 0, -2, 0, 4, 0, 0, -4, -2, -3, 0, 3], [10, 2, -2, 0, -1, 2, 0, 1, 10, 2, 0, -3], [-1, -3, 0, 1, 1, 1, -2, 0, 3, -2, -3, 0], [-2, 0, 0, 0, -3, 0, -5, -2, -5, -2, 3, 2], [-3, 1, 0, 0, 0, 1, 0, 0, -3, 1, 0, 3], [0, -5, 5, -5, -4, 0, -4, 0, 0, 0, 2, 2], [3, 0, 1, -1, 0, -4, -1, 2, -1, 1, 1, 1], [-8, 0, -2, 2, 0, -2, -5, 4, -5, -1, -1, 5], [-2, 1, -1, 2, -1, 0, 8, 2, 0, 0, -3, -3], [-1, -11, 9, -11, 4, -9, -3, -1, 3, -1, 2, 1], [-1, -3, 0, -7, -1, -7, 0, 0, 5, 0, 0, 0], [-8, 0, -3, 0, -2, 0, 0, 2, -8, -3, 0, -2], [-1, 3, 0, 3, 11, 0, -3, -6, 10, -1, 2, 1], [-5, 2, -4, 2, 3, 4, -2, -4, -2, -5, -3, 5], [1, 0, 3, -3, 0, -3, -3, 3, -6, -3, -3, 3], [5, 0, 3, -3, 0, -6, 4, -3, 5, 2, 2, -4], [1, 2, -3, 0, 0, 2, 0, 0, 1, 4, 0, -1], [0, 5, -5, 2, 7, 0, -4, 0, 0, 0, 7, 7], [3, -2, 4, -2, -5, -4, -3, 1, -2, 3, 6, -3], [4, -2, -4, -2, 4, 4, 3, -5, 8, 4, 1, -4], [6, -2, 2, 0, 1, 0, -5, -6, 0, 0, 3, 3], [-11, -3, 0, 1, -12, 1, 1, 0, -1, 1, -1, 0], [-6, 0, 1, -1, 0, 1, 1, -8, 4, -4, -4, -1], [-7, 3, 0, 1, -9, 1, 2, 0, -3, 2, 2, 0], [8, 3, -3, 9, 5, 0, 3, -8, 0, 0, 0, 0], [4, -4, 7, -4, 7, -7, 4, 3, 11, 4, 0, -4], [-12, 4, -8, 0, -7, 4, 0, 7, -12, 1, 0, 0], [3, 5, 0, 1, 1, 1, 2, 0, -7, 2, 4, 0], [-7, 3, -3, 3, 3, 3, -7, 4, -4, -7, 0, 7], [-7, -4, 3, 0, -4, -4, 0, 4, -7, -2, 0, -6], [0, 9, 0, 1, 2, 1, -2, 0, 5, -2, 1, 0], [-2, 0, 0, 0, 0, 6, -5, -4, 3, -1, -1, 5], [12, 1, -1, 0, 0, 0, 3, -12, 0, 0, -1, -1], [-6, 10, -1, 10, 5, 1, -7, 8, -1, -6, 1, 6], [-6, 7, 0, -3, -4, -3, -2, 0, -16, -2, 2, 0], [3, -4, 2, -4, -1, -2, -2, 16, 2, 3, 5, -3], [4, 1, 4, 1, -14, -4, 8, 7, -10, 4, -4, -4], [-19, -4, -1, 0, -18, -4, 0, 18, -19, 2, 0, 1], [18, 0, -1, 1, 0, -3, 1, -18, 19, 1, 1, -1], [-18, 0, -2, 2, 0, -2, 2, 4, -5, -1, -1, -2], [0, 0, -2, 2, 0, 6, -9, 3, -4, -1, -1, 9], [9, -8, 2, 0, 0, -8, 0, 0, 9, 4, 0, -4], [-7, 1, -1, 5, -17, 0, -3, 7, 0, 0, -3, -3], [0, -5, -1, -5, 0, 1, 2, 5, 0, 0, -2, 0], [5, -6, 0, -5, 4, -5, 1, 0, -16, 1, -1, 0], [5, 7, -7, 2, -3, 0, 4, -5, 0, 0, -6, -6], [8, -4, 4, -6, 1, 0, 1, -8, 0, 0, 1, 1], [12, 0, -2, 2, 0, 6, 0, -12, 14, 2, 2, 0], [9, -9, 0, -4, 7, -4, 2, 0, -3, 2, -1, 0], [4, 0, 0, 0, -12, 0, -6, -4, 0, 0, 3, 3], [22, -4, 0, -4, 23, -4, -1, 0, 0, -1, 0, 0], [4, -2, 0, 0, 15, -2, 0, -15, 4, -1, 0, -4], [-11, 8, -8, 4, -8, 0, -4, 11, 0, 0, 6, 6], [-5, 8, 0, 1, -4, 1, -1, 0, -18, -1, 1, 0], [23, 0, -3, 3, 0, 3, 6, -4, 6, 2, 2, -6], [-3, 0, 0, -9, 8, 0, -5, 3, 0, 0, 6, 6], [4, 3, -2, 3, -6, 2, 7, 9, -2, 4, -3, -4], [-16, -10, 0, -4, -15, -4, -1, 0, -2, -1, -7, 0], [-11, 1, 4, 0, 9, 1, 0, -9, -11, -2, 0, 3], [17, -1, 1, -5, 5, 0, -3, -17, 0, 0, 1, 1], [17, 0, 7, -7, 0, 0, 0, 5, 2, 7, 7, 0], [3, 0, 10, -10, 0, 1, 0, 0, 2, 2, 2, 0], [-8, 0, 2, -2, 0, 6, -2, 15, -17, -2, -2, 2], [-16, 0, -7, 7, 0, 6, 5, 3, -3, 0, 0, -5], [4, -2, 2, 10, 4, 0, 8, -4, 0, 0, -5, -5], [-2, 9, -6, 9, -14, 6, 4, 11, -16, -2, -6, 2], [5, -8, 1, -8, -10, -1, 0, 11, -5, 5, 5, -5], [-2, -5, 9, -5, -6, -9, 3, 8, -8, -2, -5, 2], [-3, -6, 6, 2, -9, 0, -1, 3, 0, 0, 0, 0], [-3, 0, 1, -1, 0, 2, 3, -12, 7, -5, -5, -3], [2, -4, 0, 4, 7, 4, -5, 0, 10, -5, 1, 0], [-11, -13, 0, -2, -6, -2, -5, 0, -2, -5, -3, 0], [-6, 3, -3, 11, -5, 0, 3, 6, 0, 0, -2, -2], [-12, 6, 4, 0, -24, 6, 0, 24, -12, 2, 0, -4], [6, 1, -2, 0, -14, 1, 0, 14, 6, -1, 0, 1], [-19, 4, 3, 0, -17, 4, 0, 17, -19, 2, 0, -4], [3, 4, -1, 4, -12, 1, 7, 22, -9, 3, -4, -3], [14, 0, 0, 2, 21, 0, -5, -14, 0, 0, 3, 3], [2, 0, -2, 2, 0, -1, 6, -6, 5, -1, -1, -6], [-8, -7, 7, -7, -4, 0, -7, 8, 0, 0, 1, 1], [4, -13, 0, -13, 5, 0, 1, -3, 9, 4, 3, -4], [23, -3, 6, 0, 6, -3, 0, -6, 23, 5, 0, -7], [-6, -5, 0, -5, 1, 0, -6, -24, -5, -6, 0, 6], [-8, -1, 5, 0, -2, -1, 0, 2, -8, -9, 0, -4], [12, 0, 0, 0, 0, -7, -2, -7, 10, 3, 3, 2], [-1, 0, -6, 6, 0, 6, 3, 6, -2, 4, 4, -3], [-6, 0, 6, -6, 0, 14, -2, 5, -5, 0, 0, 2], [-28, 3, -3, 0, 1, 3, 0, -1, -28, -7, 0, 6], [-7, 0, -5, 0, -14, 5, -5, 0, -21, -7, -2, 7], [-7, -2, 1, 0, 2, -2, 0, -2, -7, -7, 0, 1], [-7, 0, 0, -1, 2, 0, -7, 7, 0, 0, 6, 6], [-7, -15, 0, -12, -8, -12, 1, 0, 3, 1, 5, 0], [7, 5, -5, 2, 9, 0, -10, -7, 0, 0, 9, 9], [16, -2, 2, 0, 16, -2, 0, -16, 16, 3, 0, -3], [14, -11, 0, -6, 10, -6, 4, 0, 11, 4, 4, 0], [-9, 4, 1, 0, -12, 4, 0, 12, -9, 6, 0, -7], [-8, -1, -3, 0, 20, -1, 0, -20, -8, -4, 0, 1], [4, 13, 0, 8, 7, 8, -3, 0, -1, -3, 2, 0], [25, 5, -12, 0, 4, 5, 0, -4, 25, 0, 0, -2], [-5, -13, 0, -10, -12, -10, 7, 0, 3, 7, -1, 0], [18, 0, 1, -1, 0, 5, 4, -19, 21, 2, 2, -4], [-21, -4, 4, -17, -3, 0, -6, 21, 0, 0, 7, 7], [-7, 3, -3, 9, 10, 0, -6, 7, 0, 0, 4, 4], [-5, -7, 4, 0, -8, -7, 0, 8, -5, -2, 0, 3], [-6, 0, -6, 0, -12, 6, 0, -16, -18, -6, -6, 6], [0, -12, 13, -12, 1, -13, -7, 6, 1, 0, 7, 0], [9, 0, 0, -9, 13, 0, -6, -9, 0, 0, 3, 3], [12, 0, 1, -1, 0, 4, -8, -4, -1, -5, -5, 8], [11, 10, -5, 0, 17, 10, 0, -17, 11, 2, 0, 4], [1, 1, 11, 1, 7, -11, -1, 4, 8, 1, 2, -1], [11, -3, 0, 4, 0, 4, 11, 0, 20, 11, 1, 0], [-4, 10, -10, 13, -1, 0, 0, 4, 0, 0, -3, -3], [-16, 5, -5, 2, -10, 0, 9, 16, 0, 0, -11, -11], [-2, -1, 0, -9, -4, -9, 2, 0, -5, 2, 1, 0], [0, 14, 0, 1, 3, 1, -3, 0, -13, -3, 1, 0], [3, -8, 9, -8, -14, -9, 5, -10, -11, 3, -2, -3], [-12, 4, 0, -3, -6, -3, -6, 0, -12, -6, 5, 0], [19, 0, -1, 0, 25, 0, 0, -25, 19, 3, 0, 0], [7, 4, -3, 4, -4, 3, 9, 6, 3, 7, -2, -7], [13, -10, 10, -9, 10, 0, 5, -13, 0, 0, -8, -8], [15, -14, 0, -11, 16, -11, -1, 0, 24, -1, 9, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_375_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_375_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_375_2_g_c(:prec:=12)
    chi := MakeCharacter_375_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_375_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_375_2_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_375_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;