
// Make newform 3100.3.d.b in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3100_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3100_d_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_3100_3_d_b();".
// 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_3100_3_d_b();".
// 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 := [8, 0, 11, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 3, 0, 1], [0, 11, 0, 1], [6, 0, 1, 0]];
    Rf_basisdens := [1, 2, 2, 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_3100_d();"
function MakeCharacter_3100_d()
    N := 3100;
    order := 2;
    char_gens := [1551, 2977, 1801];
    v := [2, 2, 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_3100_d_Hecke();"
function MakeCharacter_3100_d_Hecke(Kf)
    N := 3100;
    order := 2;
    char_gens := [1551, 2977, 1801];
    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, 1, 0], [0, 0, 0, 0], [-2, 0, 0, 1], [0, 1, -2, 0], [0, -2, -3, 0], [0, 3, 5, 0], [0, 0, 0, -3], [0, -1, -5, 0], [0, -2, 9, 0], [15, -1, 7, -2], [0, 3, 0, 0], [-30, 0, 0, 7], [0, 0, 1, 0], [-20, 0, 0, -10], [0, 1, -16, 0], [-4, 0, 0, 9], [0, 2, -19, 0], [-62, 0, 0, -4], [4, 0, 0, -15], [0, 4, -2, 0], [0, 13, -1, 0], [0, 5, -24, 0], [0, -11, -23, 0], [-46, 0, 0, 13], [-64, 0, 0, -11], [88, 0, 0, 11], [104, 0, 0, 3], [60, 0, 0, -23], [74, 0, 0, -27], [0, 14, -10, 0], [-114, 0, 0, 4], [0, -5, -9, 0], [0, -17, -46, 0], [-18, 0, 0, -44], [0, 15, -5, 0], [-116, 0, 0, -17], [216, 0, 0, -13], [0, 12, -18, 0], [10, 0, 0, 20], [0, -4, -17, 0], [0, 8, -1, 0], [14, 0, 0, -21], [32, 0, 0, -21], [0, -11, 70, 0], [0, 28, 34, 0], [188, 0, 0, -23], [0, 38, 50, 0], [86, 0, 0, -48], [0, 9, -68, 0], [-154, 0, 0, 17], [0, 7, 41, 0], [0, -16, 12, 0], [0, 36, 23, 0], [-268, 0, 0, -1], [0, -41, -77, 0], [0, -21, -38, 0], [0, -19, -57, 0], [0, -10, 51, 0], [136, 0, 0, 29], [-66, 0, 0, 8], [230, 0, 0, -80], [392, 0, 0, 19], [312, 0, 0, -59], [0, 39, -77, 0], [56, 0, 0, 27], [0, 39, -118, 0], [0, -5, -59, 0], [0, -30, 73, 0], [-534, 0, 0, -16], [0, 16, 96, 0], [170, 0, 0, -89], [0, 24, 8, 0], [-12, 0, 0, 31], [-50, 0, 0, 52], [0, -24, -36, 0], [0, 23, -36, 0], [-80, 0, 0, -5], [0, 43, -31, 0], [0, 35, -5, 0], [84, 0, 0, -15], [-32, 0, 0, 13], [164, 0, 0, 10], [0, 47, -71, 0], [346, 0, 0, 39], [-376, 0, 0, 43], [0, -30, -70, 0], [0, -48, -146, 0], [0, 39, 182, 0], [0, 2, 74, 0], [268, 0, 0, 31], [-336, 0, 0, 109], [0, 37, 57, 0], [0, 49, 112, 0], [0, 51, -22, 0], [204, 0, 0, -97], [0, 42, 191, 0], [134, 0, 0, 20], [0, 2, -73, 0], [-576, 0, 0, -51], [160, 0, 0, -165], [0, -45, 18, 0], [580, 0, 0, -65], [0, -12, -26, 0], [0, -86, -223, 0], [-124, 0, 0, -178], [0, 39, 130, 0], [-694, 0, 0, 37], [186, 0, 0, 23], [0, -19, -127, 0], [-394, 0, 0, -88], [0, -31, -22, 0], [-278, 0, 0, 124], [0, 76, -117, 0], [0, 32, 206, 0], [0, 102, -34, 0], [0, -51, -52, 0], [0, -58, -208, 0], [-310, 0, 0, 20], [168, 0, 0, 173], [-48, 0, 0, -139], [0, 29, 243, 0], [0, 29, -32, 0], [-844, 0, 0, -33], [628, 0, 0, 73], [-892, 0, 0, 57], [0, -87, 74, 0], [0, 77, 131, 0], [-388, 0, 0, -1], [-412, 0, 0, 191], [0, -100, -45, 0], [0, 91, 199, 0], [86, 0, 0, -77], [0, -14, -91, 0], [0, -24, 38, 0], [-578, 0, 0, 51], [0, 36, -199, 0], [0, 40, -307, 0], [0, 107, 54, 0], [0, -114, 88, 0], [-502, 0, 0, 128], [0, -124, -67, 0], [0, -59, 309, 0], [0, 20, -159, 0], [0, 9, 312, 0], [-192, 0, 0, -42], [-78, 0, 0, -116], [-102, 0, 0, -84], [0, 29, -258, 0], [0, -35, -89, 0], [392, 0, 0, -61], [0, -36, 32, 0], [0, 35, 346, 0], [-810, 0, 0, -135], [-204, 0, 0, 327], [0, -95, 115, 0], [-190, 0, 0, -288], [0, 132, 236, 0], [-454, 0, 0, -148], [0, -109, -62, 0], [0, -123, 112, 0], [0, -121, 183, 0], [0, 86, 136, 0], [1102, 0, 0, -44], [-572, 0, 0, -49], [0, -142, -28, 0], [0, -21, -263, 0], [-460, 0, 0, 255], [0, -42, 214, 0], [0, 77, 74, 0], [0, 52, 121, 0], [0, 14, -263, 0], [328, 0, 0, -162], [-562, 0, 0, 121], [-1114, 0, 0, 179], [0, 94, 292, 0], [464, 0, 0, 125], [624, 0, 0, 69], [-550, 0, 0, 60], [0, 99, -8, 0], [1388, 0, 0, 31], [0, -63, 366, 0], [-1464, 0, 0, 27], [0, -20, 306, 0], [-70, 0, 0, 125], [0, 16, 193, 0], [2074, 0, 0, -12], [-224, 0, 0, -173], [0, -17, 119, 0], [-612, 0, 0, 233], [0, -93, -271, 0], [484, 0, 0, 63], [0, -70, 195, 0], [0, -52, 139, 0], [1262, 0, 0, -56], [0, 53, -83, 0], [0, -189, -127, 0], [78, 0, 0, 296], [-724, 0, 0, 167], [-1522, 0, 0, 1], [-340, 0, 0, -23], [0, -1, -3, 0], [14, 0, 0, 248], [-1378, 0, 0, 115], [-302, 0, 0, -248], [0, -130, 561, 0], [654, 0, 0, 235], [0, -227, -270, 0], [-774, 0, 0, -1], [852, 0, 0, -159], [0, 230, 66, 0], [0, -31, -418, 0], [184, 0, 0, 98], [1612, 0, 0, -81], [0, -153, -209, 0], [-142, 0, 0, -57], [-740, 0, 0, -250], [1382, 0, 0, -5], [0, -101, 303, 0], [1212, 0, 0, -161], [0, 251, 448, 0], [-92, 0, 0, 257], [-690, 0, 0, -253], [96, 0, 0, 227], [-652, 0, 0, 431], [0, -29, -402, 0], [-1044, 0, 0, -153], [0, -218, 206, 0], [0, 18, -260, 0], [2058, 0, 0, 32], [0, 29, 128, 0], [-72, 0, 0, -163], [-18, 0, 0, -389], [0, 41, 273, 0], [0, -47, 232, 0], [0, -162, -558, 0], [-1496, 0, 0, 285], [1634, 0, 0, -292], [0, 87, -194, 0], [0, 205, -295, 0], [1570, 0, 0, 60], [0, -106, -433, 0], [0, 85, 21, 0], [0, -298, -529, 0], [0, -217, -95, 0], [-1566, 0, 0, 144], [0, 99, -187, 0], [0, -271, -98, 0], [0, 149, 172, 0], [-674, 0, 0, -448], [-384, 0, 0, 107], [-1954, 0, 0, -141], [0, -115, 333, 0], [-270, 0, 0, 327], [804, 0, 0, -137], [-350, 0, 0, 232], [-552, 0, 0, -43], [0, 150, -449, 0], [-532, 0, 0, -249], [-584, 0, 0, -333], [404, 0, 0, -162], [0, 79, 520, 0], [0, 120, -205, 0], [16, 0, 0, -133], [0, 156, 416, 0], [-420, 0, 0, -159], [864, 0, 0, 229], [450, 0, 0, -224], [-894, 0, 0, 92], [140, 0, 0, -545], [874, 0, 0, -400], [0, -40, 443, 0], [0, 81, -211, 0], [0, 179, -3, 0], [-598, 0, 0, -331], [1132, 0, 0, 79], [56, 0, 0, 147], [0, -81, 402, 0], [0, -166, 152, 0], [0, 157, -344, 0], [-718, 0, 0, 149], [-2600, 0, 0, 37], [-536, 0, 0, -37], [236, 0, 0, -151], [1088, 0, 0, -229], [0, 256, -32, 0], [0, -76, -180, 0], [0, 151, 314, 0], [1056, 0, 0, -442], [0, 136, 708, 0], [-752, 0, 0, -227], [-2216, 0, 0, -117], [0, 70, 156, 0], [-146, 0, 0, 580], [0, 111, 236, 0], [0, -200, -65, 0], [0, -99, -577, 0], [2434, 0, 0, 108], [0, -89, 348, 0], [0, 107, 652, 0], [-2288, 0, 0, -86], [0, 109, -32, 0], [0, 74, -218, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3100_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3100_3_d_b();".
// 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_3100_3_d_b(:prec:=4)
    chi := MakeCharacter_3100_d();
    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_3100_3_d_b();".
// 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_3100_3_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3100_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![32, 0, 22, 0, 1]>,<7,R![-20, 3, 1]>],Snew);
    return Vf;
end function;