
// Make newform 378.3.s.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_378_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_378_s_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_378_3_s_d();".
// 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_378_3_s_d();".
// 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 := [4467, 2196, -1664, -1024, 413, 116, -34, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-4425, -1284, 1187, 192, -91, -6, 2, 0], [2711814, 4578735, -4328389, -1354204, 496373, 82598, -16681, -919], [872256, -1520571, -656684, 387688, 73885, -27020, -2534, 724], [-20444505, -7162662, 8237233, 922099, -804335, -16589, 27922, -1643], [-542487, 1269957, -527908, 262721, 130610, -45811, -6433, 1838], [-3119145, 5790408, 1313368, -775376, -147770, 54040, 5068, -1448], [-1196985, 564096, 5325251, -3006376, -801295, 291734, 28784, -8224]];
    Rf_basisdens := [1, 2418, 2749266, 1374633, 2749266, 2749266, 1374633, 2749266];
    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_378_s();"
function MakeCharacter_378_s()
    N := 378;
    order := 6;
    char_gens := [29, 325];
    v := [3, 4];
    // 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_378_s_Hecke();"
function MakeCharacter_378_s_Hecke(Kf)
    N := 378;
    order := 6;
    char_gens := [29, 325];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 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, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [-2, 2, 0, 0, 0, 0, 0, -1], [0, 0, 2, -4, 2, 2, 0, 0], [7, 0, 0, 0, 0, 0, -1, 0], [0, 0, 5, -1, 5, 5, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 12, 0, 0], [0, 0, 0, 11, 5, 11, 0, 0], [-7, -7, 0, 0, 0, 0, 0, -4], [0, -1, 0, 0, 0, 0, 9, 9], [0, 0, 0, -3, 3, -3, 0, 0], [-22, 0, 0, 0, 0, 0, 9, 0], [0, 0, -9, 0, 0, 6, 0, 0], [0, 0, 15, 21, 15, 15, 0, 0], [0, 0, 4, -44, 4, 4, 0, 0], [0, -38, 0, 0, 0, 0, 7, 7], [45, 45, 0, 0, 0, 0, 0, 7], [0, 0, 0, 64, -2, 64, 0, 0], [-66, -66, 0, 0, 0, 0, 0, 1], [0, -98, 0, 0, 0, 0, 4, 4], [0, 0, 0, -92, -8, -92, 0, 0], [0, 0, 21, 0, 0, 36, 0, 0], [6, 0, 0, 0, 0, 0, 1, 0], [0, 0, -38, -32, -38, -38, 0, 0], [0, -61, 0, 0, 0, 0, 9, 9], [0, 0, 24, 0, 0, -36, 0, 0], [138, 138, 0, 0, 0, 0, 0, 1], [0, 0, 0, 20, -10, 20, 0, 0], [34, 0, 0, 0, 0, 0, -7, 0], [0, 0, -31, 0, 0, -60, 0, 0], [0, 0, -5, -35, -5, -5, 0, 0], [51, 0, 0, 0, 0, 0, 19, 0], [0, 0, 3, 0, 0, -150, 0, 0], [81, 81, 0, 0, 0, 0, 0, -32], [-35, -35, 0, 0, 0, 0, 0, 27], [0, -97, 0, 0, 0, 0, 0, 0], [0, 0, 0, 56, -16, 56, 0, 0], [0, 0, -1, 0, 0, 144, 0, 0], [0, 0, -33, 39, -33, -33, 0, 0], [31, 0, 0, 0, 0, 0, -40, 0], [0, 0, -27, 0, 0, -54, 0, 0], [25, 25, 0, 0, 0, 0, 0, 27], [0, 0, 0, 184, 16, 184, 0, 0], [-122, -122, 0, 0, 0, 0, 0, 21], [-94, 0, 0, 0, 0, 0, -3, 0], [-85, 0, 0, 0, 0, 0, 27, 0], [0, 0, 75, 63, 75, 75, 0, 0], [0, 171, 0, 0, 0, 0, 8, 8], [0, 0, -5, 0, 0, -168, 0, 0], [0, 0, 0, 189, 21, 189, 0, 0], [22, 22, 0, 0, 0, 0, 0, 9], [0, 0, 0, -105, 39, -105, 0, 0], [0, 0, 24, 0, 0, 96, 0, 0], [0, 0, 36, -96, 36, 36, 0, 0], [0, 0, -8, -224, -8, -8, 0, 0], [0, 334, 0, 0, 0, 0, 19, 19], [70, 70, 0, 0, 0, 0, 0, 33], [0, 0, 0, 340, -14, 340, 0, 0], [102, 102, 0, 0, 0, 0, 0, -47], [0, 0, 0, 144, 54, 144, 0, 0], [-141, 0, 0, 0, 0, 0, 4, 0], [0, 0, -14, -308, -14, -14, 0, 0], [0, -105, 0, 0, 0, 0, 8, 8], [0, 0, 60, 0, 0, 144, 0, 0], [0, -274, 0, 0, 0, 0, -33, -33], [151, 0, 0, 0, 0, 0, -52, 0], [0, 0, 30, 72, 30, 30, 0, 0], [-321, 0, 0, 0, 0, 0, -20, 0], [0, 0, -48, 240, -48, -48, 0, 0], [0, 0, -64, 0, 0, -240, 0, 0], [82, 82, 0, 0, 0, 0, 0, -51], [0, -69, 0, 0, 0, 0, 80, 80], [99, 0, 0, 0, 0, 0, 55, 0], [0, 0, 46, 0, 0, 66, 0, 0], [0, 0, 61, 91, 61, 61, 0, 0], [0, 286, 0, 0, 0, 0, -29, -29], [0, 0, 24, 0, 0, -228, 0, 0], [181, 181, 0, 0, 0, 0, 0, 39], [0, 0, 0, 33, -105, 33, 0, 0], [-134, 0, 0, 0, 0, 0, 17, 0], [0, 0, -144, -108, -144, -144, 0, 0], [-466, 0, 0, 0, 0, 0, 45, 0], [0, 283, 0, 0, 0, 0, 73, 73], [0, 0, 13, 0, 0, 528, 0, 0], [0, 0, 0, -307, -103, -307, 0, 0], [0, 270, 0, 0, 0, 0, 59, 59], [0, 0, 0, -508, -40, -508, 0, 0], [-366, 0, 0, 0, 0, 0, 37, 0], [0, 0, 20, 0, 0, -120, 0, 0], [0, 0, 47, 413, 47, 47, 0, 0], [-95, -95, 0, 0, 0, 0, 0, 108], [0, 0, 0, -300, 72, -300, 0, 0], [0, -597, 0, 0, 0, 0, -28, -28], [0, 0, 0, -69, -57, -69, 0, 0], [0, 0, 30, 0, 0, 6, 0, 0], [0, 0, -12, -288, -12, -12, 0, 0], [0, 308, 0, 0, 0, 0, 102, 102], [0, -146, 0, 0, 0, 0, -125, -125], [79, 0, 0, 0, 0, 0, -124, 0], [0, 0, 103, 211, 103, 103, 0, 0], [0, 0, 11, -19, 11, 11, 0, 0], [0, 0, 30, 0, 0, 522, 0, 0], [74, 74, 0, 0, 0, 0, 0, -115], [-195, -195, 0, 0, 0, 0, 0, -17], [0, 0, 0, 43, -83, 43, 0, 0], [0, 0, 36, 0, 0, 108, 0, 0], [0, 0, 183, 201, 183, 183, 0, 0], [-502, 0, 0, 0, 0, 0, 21, 0], [0, 259, 0, 0, 0, 0, -20, -20], [722, 722, 0, 0, 0, 0, 0, -31], [0, 0, 0, -44, 22, -44, 0, 0], [753, 753, 0, 0, 0, 0, 0, -5], [-442, 0, 0, 0, 0, 0, -39, 0], [0, 0, -85, -265, -85, -85, 0, 0], [479, 0, 0, 0, 0, 0, 60, 0], [0, 0, 258, 252, 258, 258, 0, 0], [0, 0, 135, 0, 0, 252, 0, 0], [0, 0, 0, 157, 193, 157, 0, 0], [-843, -843, 0, 0, 0, 0, 0, -56], [26, 0, 0, 0, 0, 0, 45, 0], [0, 0, 78, 0, 0, 42, 0, 0], [0, 0, 186, 228, 186, 186, 0, 0], [0, -58, 0, 0, 0, 0, -117, -117], [0, 0, 0, -271, 185, -271, 0, 0], [0, 446, 0, 0, 0, 0, -9, -9], [0, 0, 72, 0, 0, -48, 0, 0], [-630, 0, 0, 0, 0, 0, -83, 0], [0, -530, 0, 0, 0, 0, 19, 19], [185, 185, 0, 0, 0, 0, 0, -100], [0, 0, 0, -651, -93, -651, 0, 0], [0, -302, 0, 0, 0, 0, 91, 91], [370, 0, 0, 0, 0, 0, -19, 0], [0, 0, -212, 0, 0, -72, 0, 0], [630, 0, 0, 0, 0, 0, -11, 0], [0, 0, -101, 391, -101, -101, 0, 0], [725, 725, 0, 0, 0, 0, 0, -40], [0, 0, 0, 12, -48, 12, 0, 0], [0, 0, 262, 148, 262, 262, 0, 0], [-105, 0, 0, 0, 0, 0, -161, 0], [0, 0, -200, 0, 0, 228, 0, 0], [198, 198, 0, 0, 0, 0, 0, 145], [0, 0, 0, -168, -246, -168, 0, 0], [-135, -135, 0, 0, 0, 0, 0, -212], [0, 0, 0, -320, -86, -320, 0, 0], [-22, 0, 0, 0, 0, 0, -27, 0], [0, 0, 18, 1092, 18, 18, 0, 0], [0, 443, 0, 0, 0, 0, -12, -12], [0, 0, 239, 0, 0, 198, 0, 0], [0, 51, 0, 0, 0, 0, -52, -52], [0, 0, 0, -279, -165, -279, 0, 0], [-973, 0, 0, 0, 0, 0, -60, 0], [0, 0, -12, 0, 0, -612, 0, 0], [1004, 1004, 0, 0, 0, 0, 0, 26], [0, 0, 0, -513, -219, -513, 0, 0], [0, 123, 0, 0, 0, 0, 164, 164], [0, 0, 33, 0, 0, 168, 0, 0], [47, 0, 0, 0, 0, 0, -69, 0], [0, 0, -146, -344, -146, -146, 0, 0], [0, 0, 173, 0, 0, 450, 0, 0], [0, 0, 0, -491, -161, -491, 0, 0], [-781, 0, 0, 0, 0, 0, -81, 0], [0, 0, -255, 0, 0, -198, 0, 0], [0, 0, -84, 108, -84, -84, 0, 0], [0, 0, 120, 288, 120, 120, 0, 0], [529, 529, 0, 0, 0, 0, 0, 231], [-1298, -1298, 0, 0, 0, 0, 0, -99]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_378_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_378_3_s_d();".
// 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_378_3_s_d(:prec:=8)
    chi := MakeCharacter_378_s();
    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_378_3_s_d();".
// 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_378_3_s_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_378_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![104976, 0, -12312, 0, 1120, 0, -38, 0, 1]>],Snew);
    return Vf;
end function;