
// Make newform 380.3.g.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_380_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_380_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_380_3_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_380_3_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 := [687241, 0, -71664, 0, 36676, 0, -270, 0, 44, 0, 20, 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], [-558303007, 0, 1825258945, 0, 54167333, 0, 4391039, 0, 524952, 0, 28618, 0], [21947164823, 0, 517724371, 0, 100053868, 0, 32340835, 0, 2443821, 0, 60512, 0], [187896630501135, -36331199656762, 2638688527393, -12940529662267, -43476949766, -162835339255, 123691327169, -28805936576, 7318528889, -5048505213, 17200092, -194997239], [-187896630501135, -36331199656762, -2638688527393, -12940529662267, 43476949766, -162835339255, -123691327169, -28805936576, -7318528889, -5048505213, -17200092, -194997239], [0, 2065201213063, 0, -354532580162, 0, -3883239581, 0, -813261775, 0, -195485288, 0, -9729001], [-104748473351889, -208524546958680, 12226184429355, 29117423544893, -4708627602976, -998868406049, -1354472427, -122950074222, 18377506607, 9572417455, 1291775986, 744934843], [-679853386340457, 208524546958680, 420227260397139, -29117423544893, -12531395244576, 998868406049, 232158453165, 122950074222, 267705610063, -9572417455, 13046909770, -744934843], [0, 25942480991957, 0, 17994962875673, 0, -565819505556, 0, 65363306369, 0, 15674641213, 0, 651724712], [0, -10974976155720, 0, 1532495976047, 0, -52572021371, 0, -6471056538, 0, 503811445, 0, 39207097], [0, 61986227495459, 0, 31467200921829, 0, -418256401478, 0, 96267253819, 0, 23103082157, 0, 1021426750], [0, -34021640801649, 0, 4527798347604, 0, -240525327873, 0, -24848763895, 0, 992670198, 0, 108734495]];
    Rf_basisdens := [1, 908842237, 1817684474, 57260696299948, 57260696299948, 1506860428946, 57260696299948, 57260696299948, 28630348149974, 1506860428946, 28630348149974, 1506860428946];
    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_380_g();"
function MakeCharacter_380_g()
    N := 380;
    order := 2;
    char_gens := [191, 77, 21];
    v := [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_380_g_Hecke();"
function MakeCharacter_380_g_Hecke(Kf)
    N := 380;
    order := 2;
    char_gens := [191, 77, 21];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, 0], [6, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0], [0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [0, -2, 0, 0, 0, 0, -2, 0, 0, 1, 0, 0], [0, 0, 0, -5, -5, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -6, -6, 0, 0, 0, -4, 0, 6, 0], [0, 0, 0, 0, 0, -7, 0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 0, 0, -3, 3, 0, 3, 0, 0], [-50, 0, 0, 7, -7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -2], [0, -4, 0, 0, 0, 0, 0, -2, 0, -1, 0, 0], [0, 0, 0, 5, 5, 0, 0, 0, -1, 0, 4, 0], [0, 6, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0], [0, 0, 0, 4, 4, 0, 0, 0, 4, 0, -8, 0], [0, -4, 0, 0, 0, 0, 4, -2, 0, -3, 0, 0], [0, 0, 0, 0, 0, 15, 0, 0, 0, -4, 0, 1], [-36, 0, -2, -15, 15, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, 0, 0, 7, 0, 1], [0, 0, 0, 0, 0, 11, 0, 0, 0, -4, 0, 0], [0, 1, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 17, 0, 0, 0, -11, 0, 1], [0, 0, 0, 0, 0, -29, 0, 0, 0, 2, 0, -1], [-56, 0, -2, -8, 8, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 3, 0, 0, 0, -3, 0, 8, 0], [-54, 0, 4, 2, -2, 0, 0, 0, 0, 0, 0, 0], [-34, 0, 8, -22, 22, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, -14, -2, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 20, 0, -10, 0], [0, 0, 0, 7, 7, 0, 0, 0, 30, 0, -10, 0], [0, 0, 0, 0, 0, 21, 0, 0, 0, 9, 0, 3], [0, 0, 0, 0, 0, 59, 0, 0, 0, 0, 0, -1], [0, -8, 0, 0, 0, 0, 10, -2, 0, -6, 0, 0], [0, -8, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0], [25, 0, -17, 30, -30, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -21, 0, 0, 0, -6, 0, -3], [0, 0, 0, -13, -13, 0, 0, 0, -22, 0, -16, 0], [9, 0, -1, 20, -20, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, -19, 3, 0, 11, 0, 0], [0, 0, 0, 0, 0, -41, 0, 0, 0, 20, 0, -1], [0, 0, 0, 0, 0, 7, 0, 0, 0, -2, 0, 0], [20, 0, 14, 34, -34, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 2, 0, 0, 0, 20, 0, -8, 0], [-75, 0, 11, 6, -6, 0, 0, 0, 0, 0, 0, 0], [0, -18, 0, 0, 0, 0, 6, -2, 0, -4, 0, 0], [86, 0, -12, 3, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 25, 0, 0, 0, 7, 0, 3], [0, 0, 0, 18, 18, 0, 0, 0, -14, 0, -14, 0], [0, 14, 0, 0, 0, 0, -20, -4, 0, 8, 0, 0], [53, 0, -1, 56, -56, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -12, -12, 0, 0, 0, 24, 0, 10, 0], [0, -16, 0, 0, 0, 0, 12, 2, 0, -5, 0, 0], [0, 0, 0, 22, 22, 0, 0, 0, 28, 0, -4, 0], [0, 0, 0, 0, 0, 9, 0, 0, 0, 19, 0, -4], [0, 0, 0, 0, 0, -1, 0, 0, 0, 6, 0, -7], [-67, 0, 23, -12, 12, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 29, 29, 0, 0, 0, 13, 0, -36, 0], [0, 0, 0, 0, 0, 5, 0, 0, 0, 11, 0, 0], [0, 13, 0, 0, 0, 0, 5, 7, 0, 1, 0, 0], [0, 0, 0, 0, 0, 29, 0, 0, 0, -5, 0, -5], [0, 0, 0, -50, -50, 0, 0, 0, 28, 0, 16, 0], [170, 0, 0, 52, -52, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 3, 0, 0, 0, -45, 0, 20, 0], [-121, 0, -31, -14, 14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 32, 32, 0, 0, 0, -44, 0, 18, 0], [0, 0, 0, 0, 0, -19, 0, 0, 0, 13, 0, -2], [0, -9, 0, 0, 0, 0, 11, 5, 0, -3, 0, 0], [0, 0, 0, 0, 0, -35, 0, 0, 0, -9, 0, 13], [56, 0, 6, -11, 11, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, 4, 0, 0, 0, 8, 0, 26, 0], [0, 8, 0, 0, 0, 0, -4, -14, 0, -5, 0, 0], [0, -6, 0, 0, 0, 0, 26, -4, 0, -15, 0, 0], [-200, 0, -26, 37, -37, 0, 0, 0, 0, 0, 0, 0], [0, 23, 0, 0, 0, 0, -5, 19, 0, 12, 0, 0], [0, -10, 0, 0, 0, 0, 22, -4, 0, -13, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0, 0, 7, 0, 11], [0, -2, 0, 0, 0, 0, -18, -28, 0, -5, 0, 0], [0, 0, 0, 93, 93, 0, 0, 0, -14, 0, -14, 0], [0, -18, 0, 0, 0, 0, 14, 18, 0, 2, 0, 0], [0, 0, 0, -23, -23, 0, 0, 0, -57, 0, -48, 0], [206, 0, 12, 59, -59, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -28, -28, 0, 0, 0, 108, 0, 2, 0], [0, 0, 0, 13, 13, 0, 0, 0, -10, 0, 22, 0], [146, 0, -28, -60, 60, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -25, 0, 0, 0, -2, 0, -5], [384, 0, -34, -55, 55, 0, 0, 0, 0, 0, 0, 0], [-80, 0, -30, 83, -83, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 26, 26, 0, 0, 0, -73, 0, 36, 0], [0, 4, 0, 0, 0, 0, -6, -14, 0, -4, 0, 0], [0, -2, 0, 0, 0, 0, 14, 6, 0, -4, 0, 0], [0, 0, 0, 0, 0, -125, 0, 0, 0, 6, 0, 4], [106, 0, 12, -27, 27, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 111, 0, 0, 0, 0, 0, -9], [0, 0, 0, 60, 60, 0, 0, 0, -30, 0, -22, 0], [0, 0, 0, 0, 0, -5, 0, 0, 0, -14, 0, 9], [0, -34, 0, 0, 0, 0, 10, -2, 0, -6, 0, 0], [358, 0, 24, 31, -31, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, -5, 0, 0, 0, 9, 0, 40, 0], [0, 0, 0, -21, -21, 0, 0, 0, -76, 0, 14, 0], [0, 0, 0, -22, -22, 0, 0, 0, 42, 0, -4, 0], [0, -22, 0, 0, 0, 0, -6, 4, 0, 5, 0, 0], [0, 14, 0, 0, 0, 0, 22, -20, 0, -21, 0, 0], [0, 0, 0, 0, 0, -193, 0, 0, 0, -10, 0, 9], [0, 0, 0, -57, -57, 0, 0, 0, 64, 0, 24, 0], [0, 0, 0, -18, -18, 0, 0, 0, 76, 0, 20, 0], [128, 0, 2, 11, -11, 0, 0, 0, 0, 0, 0, 0], [-144, 0, 34, 56, -56, 0, 0, 0, 0, 0, 0, 0], [0, 28, 0, 0, 0, 0, -36, 30, 0, 33, 0, 0], [0, 0, 0, 62, 62, 0, 0, 0, -54, 0, 20, 0], [0, 0, 0, 20, 20, 0, 0, 0, 55, 0, -12, 0], [0, 0, 0, -2, -2, 0, 0, 0, 80, 0, -38, 0], [0, -31, 0, 0, 0, 0, -15, -25, 0, -5, 0, 0], [0, 1, 0, 0, 0, 0, 1, 21, 0, 10, 0, 0], [0, 0, 0, 0, 0, 131, 0, 0, 0, -28, 0, -3], [0, 0, 0, 0, 0, 1, 0, 0, 0, 47, 0, -4], [0, 0, 0, 0, 0, 75, 0, 0, 0, 54, 0, -3], [284, 0, -10, 94, -94, 0, 0, 0, 0, 0, 0, 0], [96, 0, 58, -64, 64, 0, 0, 0, 0, 0, 0, 0], [-216, 0, -34, -28, 28, 0, 0, 0, 0, 0, 0, 0], [-589, 0, -67, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -78, -78, 0, 0, 0, 135, 0, -12, 0], [0, 0, 0, 88, 88, 0, 0, 0, -6, 0, -58, 0], [456, 0, -58, -37, 37, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 129, 0, 0, 0, 15, 0, -21], [0, 16, 0, 0, 0, 0, 12, 10, 0, -1, 0, 0], [0, 0, 0, 44, 44, 0, 0, 0, 10, 0, 22, 0], [-119, 0, 27, 103, -103, 0, 0, 0, 0, 0, 0, 0], [317, 0, 47, -103, 103, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 185, 0, 0, 0, -39, 0, 2], [0, 0, 0, 0, 0, 71, 0, 0, 0, -40, 0, -6], [0, 0, 0, 0, 0, -135, 0, 0, 0, 47, 0, -8], [325, 0, 19, -57, 57, 0, 0, 0, 0, 0, 0, 0], [0, -29, 0, 0, 0, 0, 35, 41, 0, 3, 0, 0], [310, 0, 28, -40, 40, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -72, -72, 0, 0, 0, -65, 0, 88, 0], [0, 0, 0, 0, 0, -69, 0, 0, 0, -62, 0, -16], [0, -45, 0, 0, 0, 0, 35, -33, 0, -34, 0, 0], [0, -56, 0, 0, 0, 0, 8, 16, 0, 4, 0, 0], [0, 0, 0, 106, 106, 0, 0, 0, -70, 0, -30, 0], [0, 0, 0, 0, 0, -69, 0, 0, 0, -18, 0, 15], [152, 0, 54, -51, 51, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 253, 0, 0, 0, -63, 0, 1], [0, 0, 0, 0, 0, 229, 0, 0, 0, -51, 0, 16], [284, 0, -82, -24, 24, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 68, 68, 0, 0, 0, 84, 0, -40, 0], [0, 0, 0, 0, 0, 13, 0, 0, 0, 23, 0, -7], [0, 0, 0, 0, 0, 139, 0, 0, 0, 40, 0, -6], [0, -40, 0, 0, 0, 0, 64, 8, 0, -28, 0, 0], [-135, 0, 47, 64, -64, 0, 0, 0, 0, 0, 0, 0], [-935, 0, 55, -29, 29, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 35, 35, 0, 0, 0, -231, 0, 24, 0], [0, 1, 0, 0, 0, 0, 41, -27, 0, -34, 0, 0], [0, 0, 0, 42, 42, 0, 0, 0, -20, 0, 80, 0], [0, 0, 0, 0, 0, -127, 0, 0, 0, 19, 0, 19], [0, 0, 0, 20, 20, 0, 0, 0, 84, 0, -38, 0], [0, 22, 0, 0, 0, 0, -24, -60, 0, -18, 0, 0], [0, 0, 0, 0, 0, -219, 0, 0, 0, 23, 0, -11], [0, 0, 0, 0, 0, -317, 0, 0, 0, -12, 0, 7], [0, -26, 0, 0, 0, 0, -34, -8, 0, 13, 0, 0], [0, 0, 0, 5, 5, 0, 0, 0, -186, 0, -36, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_380_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_380_3_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_380_3_g_c(:prec:=12)
    chi := MakeCharacter_380_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_380_3_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_380_3_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_380_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![-1280, 0, 497, 0, -46, 0, 1]>],Snew);
    return Vf;
end function;