
// Make newform 2880.3.e.e in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2880_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2880_e_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_2880_3_e_e();".
// 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_2880_3_e_e();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 2], [1, 0, 2, -2], [-4, 8, -4, 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_2880_e();"
function MakeCharacter_2880_e()
    N := 2880;
    order := 2;
    char_gens := [2431, 901, 641, 577];
    v := [1, 2, 2, 2];
    // 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_2880_e_Hecke();"
function MakeCharacter_2880_e_Hecke(Kf)
    N := 2880;
    order := 2;
    char_gens := [2431, 901, 641, 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, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, -1], [0, -2, 0, 2], [4, 0, -2, 0], [6, 0, 8, 0], [0, 0, 0, 2], [0, 3, 0, 1], [-2, 0, 4, 0], [0, 10, 0, 2], [-4, 0, -10, 0], [28, 0, 6, 0], [0, 3, 0, -2], [0, 13, 0, -3], [-44, 0, 10, 0], [0, 12, 0, 6], [-32, 0, 26, 0], [0, -11, 0, -10], [0, -2, 0, 10], [66, 0, 32, 0], [0, -20, 0, -8], [0, 13, 0, -6], [22, 0, -40, 0], [-66, 0, 12, 0], [82, 0, 8, 0], [0, -23, 0, 11], [0, -17, 0, -12], [32, 0, 74, 0], [-126, 0, 20, 0], [0, 23, 0, -19], [0, 42, 0, 14], [26, 0, 12, 0], [0, -4, 0, 30], [48, 0, 38, 0], [0, -62, 0, 18], [-84, 0, 54, 0], [0, -45, 0, -28], [0, -15, 0, -9], [-204, 0, 10, 0], [0, -76, 0, 6], [-122, 0, -16, 0], [0, 58, 0, 22], [146, 0, 16, 0], [-16, 0, 70, 0], [0, 64, 0, -16], [0, -42, 0, 14], [0, 13, 0, 35], [0, -1, 0, -8], [-118, 0, 20, 0], [-6, 0, 140, 0], [0, 36, 0, -20], [-68, 0, -30, 0], [0, -26, 0, -26], [266, 0, -4, 0], [0, 79, 0, -29], [-192, 0, 70, 0], [0, 86, 0, -6], [256, 0, 90, 0], [28, 0, -10, 0], [0, 15, 0, 24], [76, 0, -46, 0], [0, -11, 0, 34], [0, 10, 0, -34], [-174, 0, 176, 0], [264, 0, 46, 0], [0, 70, 0, -22], [-126, 0, 64, 0], [0, 91, 0, -38], [242, 0, 108, 0], [14, 0, -144, 0], [0, 16, 0, -8], [0, -157, 0, 17], [-256, 0, 154, 0], [0, 32, 0, -30], [0, 43, 0, 57], [48, 0, -154, 0], [-124, 0, 94, 0], [-282, 0, -72, 0], [-272, 0, 150, 0], [0, -28, 0, -70], [328, 0, 114, 0], [0, -126, 0, 22], [-234, 0, -60, 0], [0, 48, 0, 64], [0, -19, 0, -44], [-528, 0, -102, 0], [554, 0, -120, 0], [-278, 0, 80, 0], [0, -167, 0, 1], [0, -13, 0, 116], [0, -172, 0, 16], [0, 67, 0, 79], [0, 38, 0, -26], [0, -24, 0, -114], [0, -73, 0, -107], [-602, 0, 140, 0], [-202, 0, 360, 0], [0, -161, 0, 52], [438, 0, 80, 0], [0, 1, 0, 20], [704, 0, 86, 0], [0, 89, 0, 92], [112, 0, -54, 0], [0, -122, 0, -10], [-186, 0, -236, 0], [0, -153, 0, 34], [94, 0, 96, 0], [0, -12, 0, 108], [172, 0, 82, 0], [0, 251, 0, -69], [-656, 0, -70, 0], [-54, 0, 220, 0], [0, 200, 0, 2], [0, -198, 0, -26], [-212, 0, -282, 0], [0, 271, 0, 26], [0, 117, 0, -19], [-464, 0, -42, 0], [0, 92, 0, -14], [48, 0, -342, 0], [-114, 0, -260, 0], [-136, 0, -306, 0], [0, 61, 0, 22], [0, -86, 0, -2], [-568, 0, 62, 0], [-518, 0, -316, 0], [0, -76, 0, -44], [0, 55, 0, -77], [-236, 0, 446, 0], [0, 36, 0, 10], [0, -213, 0, 67], [0, 282, 0, -58], [576, 0, -134, 0], [-122, 0, -344, 0], [-102, 0, -328, 0], [-104, 0, 142, 0], [0, 169, 0, -58], [744, 0, -210, 0], [-218, 0, -200, 0], [0, 122, 0, -22], [232, 0, -402, 0], [0, -115, 0, 67], [0, 267, 0, 110], [352, 0, -70, 0], [0, 220, 0, 104], [264, 0, -590, 0], [426, 0, -260, 0], [0, -52, 0, -6], [0, 147, 0, 79], [-284, 0, 142, 0], [348, 0, 262, 0], [0, 103, 0, 32], [0, -199, 0, 9], [0, -27, 0, -52], [0, -442, 0, 54], [0, -204, 0, 44], [-968, 0, -254, 0], [294, 0, 20, 0], [1082, 0, 160, 0], [0, 259, 0, -70], [-866, 0, 192, 0], [0, -125, 0, -91], [0, -246, 0, -198], [1226, 0, -68, 0], [0, -65, 0, -61], [0, -14, 0, 122], [316, 0, -346, 0], [-502, 0, -504, 0], [-1424, 0, -218, 0], [0, -96, 0, 22], [238, 0, -440, 0], [0, -158, 0, -270], [426, 0, 328, 0], [0, -160, 0, -204], [-578, 0, 528, 0], [0, 278, 0, -62], [-158, 0, -200, 0], [0, -75, 0, 103], [-768, 0, 106, 0], [0, 307, 0, -11], [0, 66, 0, 202], [-156, 0, -754, 0], [1166, 0, 48, 0], [0, -101, 0, 55], [-642, 0, -12, 0], [576, 0, 458, 0], [0, -433, 0, -30], [-72, 0, 94, 0], [0, 214, 0, 234], [326, 0, -236, 0], [0, 37, 0, -260], [0, -62, 0, -70], [-528, 0, -410, 0], [0, -329, 0, 18], [-1086, 0, 84, 0], [-668, 0, 410, 0], [-776, 0, -142, 0], [1006, 0, -240, 0], [0, 431, 0, 49], [-1082, 0, -180, 0], [0, 246, 0, 78], [-4, 0, -154, 0], [-502, 0, -552, 0], [0, 396, 0, 90], [324, 0, 890, 0], [0, -72, 0, 60], [0, -455, 0, 26], [-418, 0, -496, 0], [0, -34, 0, 170], [674, 0, 480, 0], [632, 0, -50, 0], [0, 1, 0, -115], [0, -265, 0, 130], [0, -332, 0, -60], [412, 0, 370, 0], [0, -113, 0, 265], [-1042, 0, -544, 0], [0, -255, 0, -209], [1476, 0, -150, 0], [368, 0, 554, 0], [0, 4, 0, -256], [-498, 0, -448, 0], [0, 457, 0, -81], [0, -273, 0, -70], [672, 0, 186, 0], [-546, 0, 0, 0], [0, 152, 0, -24], [0, -237, 0, 69], [0, -374, 0, 86], [1284, 0, 222, 0], [0, 688, 0, -86], [0, 238, 0, 206], [108, 0, -666, 0], [0, 319, 0, -202], [0, -183, 0, 275], [-1432, 0, 350, 0], [-1244, 0, 26, 0], [0, 824, 0, -66], [0, 266, 0, 322], [0, 629, 0, 6], [0, -22, 0, -134], [0, -67, 0, 273], [-838, 0, 212, 0], [734, 0, 0, 0], [0, -168, 0, 128], [0, -353, 0, -339], [0, 222, 0, 170], [0, -348, 0, -110], [0, -313, 0, -229], [2196, 0, -82, 0], [-762, 0, 472, 0], [0, -3, 0, 217], [1008, 0, -154, 0], [-484, 0, -174, 0], [0, 712, 0, 30], [1078, 0, -96, 0], [0, -547, 0, 10], [1744, 0, -10, 0], [-126, 0, 96, 0], [0, -403, 0, -135], [0, -169, 0, -180], [-198, 0, -380, 0], [-2856, 0, 194, 0], [-354, 0, 608, 0], [0, 440, 0, -50], [318, 0, 1316, 0], [-922, 0, 504, 0], [0, -61, 0, 12], [476, 0, 770, 0], [-2002, 0, 456, 0], [0, -483, 0, -244], [-1314, 0, -660, 0], [0, -864, 0, 0], [1374, 0, 300, 0], [0, 117, 0, -67], [0, -37, 0, -220], [-1558, 0, -60, 0], [1372, 0, 258, 0], [0, -62, 0, -158], [0, 283, 0, 211], [0, -194, 0, -242], [0, 589, 0, 3], [2838, 0, -64, 0], [0, -287, 0, 354], [0, -170, 0, 62], [-1014, 0, 712, 0], [1784, 0, 590, 0], [0, -44, 0, 408], [1032, 0, 674, 0], [312, 0, -962, 0], [0, 95, 0, 510], [-186, 0, 520, 0], [0, -226, 0, 46], [1124, 0, 590, 0], [-912, 0, -842, 0], [0, 170, 0, -190], [-1764, 0, 130, 0], [0, -808, 0, 234], [0, 389, 0, -148], [-314, 0, 1236, 0], [-676, 0, 882, 0], [0, 340, 0, 80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2880_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2880_3_e_e();".
// 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_2880_3_e_e(:prec:=4)
    chi := MakeCharacter_2880_e();
    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_2880_3_e_e();".
// 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_2880_3_e_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2880_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![2000, 0, 100, 0, 1]>,<13,R![-4, -8, 1]>,<17,R![-284, -12, 1]>],Snew);
    return Vf;
end function;