
// Make newform 6192.2.a.t in Magma, downloaded from the LMFDB on 15 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_6192_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6192_2_a_t();".
// 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_6192_2_a_t();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6192_a();"
function MakeCharacter_6192_a()
    N := 6192;
    order := 1;
    char_gens := [3871, 4645, 4817, 433];
    v := [1, 1, 1, 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;

function MakeCharacter_6192_a_Hecke( : Kf := Kf)
    return MakeCharacter_6192_a();
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 := 2;
    raw_aps := [[0], [0], [2], [-2], [0], [2], [-6], [-4], [6], [2], [-4], [4], [2], [-1], [6], [4], [-8], [-12], [-4], [0], [-14], [-8], [4], [-10], [-2], [12], [-12], [20], [10], [-18], [0], [-12], [10], [20], [-18], [-2], [-24], [0], [-6], [12], [4], [6], [-8], [14], [-12], [-14], [16], [14], [-28], [-10], [6], [10], [-10], [-24], [18], [-12], [-4], [-4], [12], [22], [4], [-4], [-28], [-26], [14], [-8], [-16], [-26], [4], [24], [-2], [34], [-32], [4], [20], [-16], [14], [-26], [10], [-26], [28], [20], [-22], [38], [-28], [-36], [6], [22], [-12], [38], [-12], [-6], [12], [-20], [-28], [12], [16], [26], [28], [38], [12], [-12], [-12], [42], [-24], [-22], [-28], [-18], [22], [-30], [-46], [-46], [-18], [-20], [38], [-30], [36], [-12], [-26], [8], [46], [-10], [-22], [-40], [-8], [36], [-38], [22], [2], [36], [12], [36], [-14], [-48], [-26], [30], [-46], [-4], [-20], [26], [-44], [24], [-16], [48], [4], [12], [-42], [42], [48], [0], [10], [-10], [-4], [8], [4], [12], [52], [34], [50], [36], [40], [30], [-12], [-12], [-6], [20], [-22], [16], [-2], [-24], [44], [44], [16], [26], [10], [62], [-40], [30], [60], [28], [-22], [-12], [-28], [50], [-60], [54], [40], [-32], [22], [-12], [34], [36], [-60], [-34], [36], [50], [34], [-26], [22], [-24], [-16], [-34], [-52], [50], [-12], [-22], [-30], [44], [-46], [4], [-2], [16], [40], [0], [4], [26], [-22], [18], [-24], [-48], [-8], [28], [14], [4], [4], [2], [-34], [12], [62], [28], [-12], [52], [56], [-6], [-12], [-66], [14], [-36], [4], [-42], [44], [-28], [8], [-54], [2], [42], [38], [48], [-28], [50], [46], [14], [46], [-14], [-2], [-36], [4], [-28], [66], [18], [-46], [36], [-14], [-6], [2], [36], [6], [-26], [-16], [-44], [-62], [28], [-30], [14], [38], [58], [32], [-8], [70], [12], [-46], [4], [-42], [44], [-8], [0], [-14], [34], [62], [-2], [24], [60], [6], [12], [-26], [-84], [68], [36], [-84], [-52], [-78], [-46], [-40], [-60], [-16], [-58], [12], [16], [64], [44], [20], [-50], [74], [-80], [-42], [-2], [-4], [-82], [14], [18], [20], [-58], [-46], [16], [-26], [-34], [8], [-28], [-18], [-34], [-68], [-88], [2], [-12], [68], [48], [4], [-2], [-62], [38], [30], [-58], [54], [2], [72], [24], [-24], [12], [0], [8], [76], [62], [44], [-74], [-18], [-42], [-4], [20], [-6], [-30], [32], [-46], [36], [60], [36], [-94], [-82], [-44], [46], [-36], [-56], [50], [90], [-60], [20], [60], [-42], [46], [-10], [-26], [-32], [-22], [36], [-98], [-52], [102], [-102], [22], [52], [90], [-58], [-26], [-20], [-76], [-4], [-22], [56], [26], [-32], [62], [-70], [66], [-28], [-102], [-6], [30], [8], [10], [4], [16], [70], [2], [-44], [4], [-6], [-16], [30], [88], [58], [-76], [-78], [-26], [36], [-4], [-2], [-58], [-76], [70], [52], [-84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6192_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6192_2_a_t();".
// 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_6192_2_a_t(:prec:=1)
    chi := MakeCharacter_6192_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6192_2_a_t();".
// 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_6192_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6192_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![2, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;