
// Make newform 2304.3.b.k in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_b_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_2304_3_b_k();".
// 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_2304_3_b_k();".
// 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, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 2], [0, 8, 0, -4], [-4, 0, 8, 0]];
    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_2304_b();"
function MakeCharacter_2304_b()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [1, 1, 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_2304_b_Hecke();"
function MakeCharacter_2304_b_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    char_values := [[-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, 1, 0, -1], [0, -2, 0, 1], [-8, 0, -1, 0], [0, -5, 0, -2], [6, 0, 2, 0], [24, 0, 1, 0], [0, 4, 0, 0], [0, -5, 0, -1], [0, 2, 0, 5], [0, 9, 0, -4], [-22, 0, 2, 0], [-40, 0, 5, 0], [0, -28, 0, -2], [0, -3, 0, 1], [-32, 0, -11, 0], [0, 7, 0, 0], [-32, 0, 7, 0], [0, 20, 0, -12], [30, 0, -8, 0], [0, -38, 0, -3], [-56, 0, -1, 0], [-78, 0, 4, 0], [-62, 0, -12, 0], [0, -83, 0, 1], [0, 2, 0, 3], [-48, 0, 13, 0], [0, 3, 0, 2], [-82, 0, 12, 0], [0, 6, 0, 23], [-64, 0, 1, 0], [26, 0, -26, 0], [64, 0, 15, 0], [0, -31, 0, -9], [0, 14, 0, -11], [0, -97, 0, 4], [72, 0, -15, 0], [0, -112, 0, 2], [0, 23, 0, 17], [96, 0, 13, 0], [0, -27, 0, 14], [0, -16, 0, -8], [82, 0, -8, 0], [0, 53, 0, 21], [0, -86, 0, 15], [-224, 0, -5, 0], [0, -34, 0, 11], [-232, 0, -1, 0], [0, 13, 0, -30], [-302, 0, -12, 0], [0, 8, 0, 56], [-190, 0, 20, 0], [-72, 0, 11, 0], [-130, 0, 32, 0], [0, 128, 0, 2], [0, -69, 0, -9], [0, -162, 0, -25], [0, -11, 0, -34], [194, 0, 0, 0], [256, 0, 11, 0], [0, 13, 0, -71], [96, 0, -45, 0], [0, 136, 0, -14], [46, 0, 56, 0], [0, -77, 0, 35], [-80, 0, 27, 0], [-94, 0, 8, 0], [-232, 0, 43, 0], [0, -73, 0, -32], [-242, 0, 12, 0], [0, 84, 0, -4], [0, -62, 0, -55], [0, -31, 0, 40], [-184, 0, -3, 0], [0, -128, 0, -48], [0, 49, 0, 63], [0, 87, 0, 24], [-74, 0, 54, 0], [-98, 0, -60, 0], [376, 0, -41, 0], [0, -139, 0, -34], [0, -36, 0, -54], [210, 0, -24, 0], [0, 194, 0, -1], [-424, 0, 35, 0], [198, 0, -86, 0], [-610, 0, 12, 0], [0, 247, 0, 57], [0, 206, 0, 15], [-72, 0, -53, 0], [0, 28, 0, -34], [0, -306, 0, 17], [16, 0, -35, 0], [128, 0, -53, 0], [0, 100, 0, -72], [0, -173, 0, -53], [458, 0, 42, 0], [-56, 0, 57, 0], [0, 19, 0, -6], [440, 0, 13, 0], [0, 399, 0, -11], [-56, 0, 7, 0], [-182, 0, -46, 0], [560, 0, -1, 0], [834, 0, 8, 0], [368, 0, -99, 0], [-18, 0, 76, 0], [0, -100, 0, 108], [542, 0, 0, 0], [0, 406, 0, -41], [0, 369, 0, 0], [2, 0, 56, 0], [-224, 0, -113, 0], [0, 366, 0, 45], [422, 0, 98, 0], [-488, 0, -3, 0], [0, -444, 0, 12], [0, 431, 0, 61], [656, 0, -63, 0], [0, -119, 0, -52], [-686, 0, 16, 0], [0, 409, 0, -53], [168, 0, 147, 0], [-1016, 0, 21, 0], [0, 99, 0, 3], [0, 61, 0, -30], [0, -540, 0, 38], [0, 14, 0, 5], [0, 331, 0, -106], [464, 0, 27, 0], [0, 344, 0, 38], [0, -182, 0, 93], [0, 61, 0, 18], [698, 0, 6, 0], [242, 0, 112, 0], [0, -75, 0, 29], [-184, 0, -83, 0], [0, 31, 0, -11], [-102, 0, 118, 0], [216, 0, 5, 0], [0, 437, 0, 5], [0, -122, 0, -39], [624, 0, 89, 0], [0, 3, 0, 50], [0, 76, 0, 128], [0, -151, 0, -156], [-118, 0, 146, 0], [-88, 0, -211, 0], [0, -48, 0, 64], [0, -137, 0, -136], [-450, 0, 104, 0], [168, 0, 29, 0], [0, -224, 0, 30], [-760, 0, -55, 0], [0, -296, 0, -168], [0, -378, 0, -95], [310, 0, 222, 0], [-162, 0, 48, 0], [0, 359, 0, -63], [224, 0, 189, 0], [234, 0, 26, 0], [0, -14, 0, -29], [8, 0, -141, 0], [662, 0, -2, 0], [0, -400, 0, -34], [0, 90, 0, -175], [0, 145, 0, 24], [1026, 0, -36, 0], [0, -631, 0, 19], [-1168, 0, -39, 0], [0, 127, 0, -132], [0, -104, 0, -18], [-994, 0, -128, 0], [0, -62, 0, 145], [-62, 0, -96, 0], [128, 0, 123, 0], [0, 81, 0, -33], [0, 498, 0, -53], [0, 143, 0, -172], [0, 554, 0, -111], [1384, 0, -69, 0], [0, -227, 0, 10], [1122, 0, -48, 0], [0, -320, 0, 44], [0, -199, 0, -101], [0, -85, 0, 30], [-32, 0, 111, 0], [190, 0, -52, 0], [0, 96, 0, -80], [-222, 0, -88, 0], [-320, 0, -143, 0], [-192, 0, 243, 0], [0, 431, 0, -75], [192, 0, 73, 0], [-830, 0, -152, 0], [418, 0, 120, 0], [0, -565, 0, 126], [-418, 0, 184, 0], [0, 312, 0, 94], [0, 23, 0, 161], [0, 110, 0, 223], [0, -491, 0, -146], [-1342, 0, -104, 0], [168, 0, 3, 0], [0, -381, 0, -197], [0, -522, 0, -113], [-1432, 0, 27, 0], [1330, 0, 76, 0], [-440, 0, 201, 0], [210, 0, 176, 0], [0, -343, 0, 195], [0, 82, 0, 79], [1408, 0, 69, 0], [0, -628, 0, -168], [-914, 0, -88, 0], [0, -118, 0, -115], [646, 0, -110, 0], [0, -912, 0, 46], [0, -569, 0, -39], [0, 389, 0, -74], [0, 818, 0, 103], [1846, 0, -98, 0], [0, 222, 0, -153], [-376, 0, -177, 0], [0, -839, 0, 108], [466, 0, 316, 0], [0, -36, 0, 206], [0, -310, 0, -249], [1552, 0, -115, 0], [0, -637, 0, -110], [-1224, 0, -159, 0], [0, 302, 0, 203], [-254, 0, -56, 0], [136, 0, 153, 0], [0, 716, 0, -22], [-1406, 0, 44, 0], [0, 33, 0, -185], [-440, 0, 71, 0], [0, -540, 0, -172], [1672, 0, 23, 0], [-528, 0, 239, 0], [0, -274, 0, -135], [0, 63, 0, -356], [-1826, 0, 32, 0], [0, 796, 0, 12], [0, 498, 0, 253], [-656, 0, -187, 0], [-1360, 0, -109, 0], [0, -552, 0, 160], [0, 403, 0, 194], [1998, 0, -84, 0], [0, -140, 0, 308], [-194, 0, 360, 0], [0, 59, 0, -233], [2416, 0, -39, 0], [0, 1317, 0, -2], [1456, 0, -89, 0], [0, -427, 0, -51], [1614, 0, 200, 0], [0, 550, 0, -89], [-2336, 0, 137, 0], [0, 853, 0, 214], [0, 543, 0, 92], [-938, 0, -26, 0], [368, 0, -237, 0], [0, -81, 0, 141], [82, 0, 252, 0], [-1096, 0, -167, 0], [0, -655, 0, 223], [0, 679, 0, -184], [624, 0, -321, 0], [-2, 0, -352, 0], [0, -518, 0, -159], [-350, 0, 32, 0], [0, -606, 0, 31], [328, 0, -1, 0], [0, -281, 0, -112], [-642, 0, 16, 0], [-784, 0, -39, 0], [0, -356, 0, 110], [0, 1394, 0, -53], [0, 332, 0, 380], [0, 385, 0, -408], [88, 0, 109, 0], [0, 328, 0, 240], [-222, 0, 12, 0], [0, -695, 0, 171], [0, 190, 0, -419], [14, 0, 12, 0], [0, -401, 0, 237], [-1472, 0, -219, 0], [-1646, 0, 188, 0], [704, 0, -31, 0], [0, -1537, 0, 36], [0, -121, 0, 129], [0, -314, 0, 287], [0, -547, 0, -71], [-408, 0, 343, 0], [-2648, 0, -47, 0], [622, 0, 96, 0], [0, -897, 0, -259], [0, 114, 0, -199]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_b_k();".
// 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_2304_3_b_k(:prec:=4)
    chi := MakeCharacter_2304_b();
    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_2304_3_b_k();".
// 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_2304_3_b_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![1936, 0, 104, 0, 1]>,<7,R![1024, 0, 128, 0, 1]>,<11,R![16, 16, 1]>,<17,R![-156, -12, 1]>,<19,R![528, -48, 1]>],Snew);
    return Vf;
end function;