
// Make newform 350.6.c.g in Magma, downloaded from the LMFDB on 03 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_c_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_350_6_c_g();".
// 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_350_6_c_g();".
// 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];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_350_c();"
function MakeCharacter_350_c()
    N := 350;
    order := 2;
    char_gens := [127, 101];
    v := [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_350_c_Hecke();"
function MakeCharacter_350_c_Hecke( : Kf := false)
    N := 350;
    order := 2;
    char_gens := [127, 101];
    char_values := [[-1, 0], [1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 6;
    raw_aps := [[0, -4], [0, 9], [0, 0], [0, -49], [-187, 0], [0, -627], [0, 1813], [-258, 0], [0, -2970], [-1299, 0], [1916, 0], [0, 6578], [6676, 0], [0, -3178], [0, -22001], [0, -26168], [-3932, 0], [-48740, 0], [0, -44832], [63736, 0], [0, -60470], [43721, 0], [0, -97276], [-45560, 0], [0, -57295], [-44970, 0], [0, 101405], [0, -166002], [-8289, 0], [0, -263206], [0, 30052], [-120050, 0], [0, 31776], [200162, 0], [-309642, 0], [-208657, 0], [0, 36010], [0, -175670], [0, -157413], [0, 23471], [-612228, 0], [528832, 0], [540369, 0], [0, -960320], [0, 761944], [-125084, 0], [627547, 0], [0, 1221101], [0, 390547], [712124, 0], [0, -561576], [1360837, 0], [530050, 0], [990330, 0], [0, -1816430], [0, 1958474], [-218034, 0], [1262652, 0], [0, -1102638], [-998213, 0], [0, 386371], [0, 783571], [0, 2817733], [-847398, 0], [0, -364955], [0, 1937438], [-61460, 0], [0, -3741158], [0, 211334], [-3395578, 0], [0, 3880935], [-3242104, 0], [0, 1444303], [0, -3435416], [1686348, 0], [0, -2643540], [452099, 0], [0, -1955303], [-4767367, 0], [4131990, 0], [-190512, 0], [-5191855, 0], [-4217811, 0], [0, 4860274], [2031134, 0], [0, -2841986], [4596825, 0], [0, 4933672], [-4756672, 0], [0, -4086188], [0, -4159319], [3360402, 0], [0, 7052426], [-83937, 0], [7105263, 0], [0, -2891469], [-1035482, 0], [-7497146, 0], [0, -3532228], [-4991877, 0], [0, 5126340], [0, 8868656], [0, -9072772], [-2083098, 0], [-5463684, 0], [0, 7662463], [0, -15746456], [0, 16240857], [1907931, 0], [3529698, 0], [0, 3377989], [0, -1204122], [0, -5473298], [-3226622, 0], [13628237, 0], [-19247150, 0], [0, 12839945], [0, 20023256], [0, 7236554], [-14247449, 0], [14926480, 0], [0, 15506168], [0, -7800645], [0, -15854668], [20365636, 0], [24803615, 0], [-18191707, 0], [16620210, 0], [0, -15759144], [0, 2152385], [22726715, 0], [0, -12115296], [-20759041, 0], [0, -18622184], [26533626, 0], [20159462, 0], [0, 5864877], [0, -1633469], [0, 20767305], [-3539361, 0], [-21148006, 0], [265389, 0], [0, -30926116], [0, -28415246], [33354724, 0], [5662046, 0], [0, -21998330], [0, -21756810], [40938396, 0], [0, -5655968], [0, 26106662], [1442944, 0], [0, 15243168], [0, -33149984], [0, 11696306], [28932096, 0], [45783829, 0], [-24694712, 0], [0, -19892551], [-37345436, 0], [0, 51039592], [0, -254832], [0, 11701186], [35907984, 0], [0, -55059198], [0, 18131723], [-20290760, 0], [0, 47539037]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_6_c_g();".
// 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_350_6_c_g(:prec:=2)
    chi := MakeCharacter_350_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_350_6_c_g();".
// 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_350_6_c_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![81, 0, 1]>,<11,R![187, 1]>],Snew);
    return Vf;
end function;