
// Make newform 345.2.a.i in Magma, downloaded from the LMFDB on 13 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_345_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_345_2_a_i();".
// 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_345_2_a_i();".
// 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 := [-6, 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_345_a();"
function MakeCharacter_345_a()
    N := 345;
    order := 1;
    char_gens := [116, 277, 166];
    v := [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_345_a_Hecke( : Kf := Kf)
    return MakeCharacter_345_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, 1], [1, 0], [-1, 0], [-1, 0], [0, -1], [2, -1], [-3, 1], [2, -1], [-1, 0], [3, -3], [5, 2], [-1, -2], [3, -1], [2, 0], [6, -1], [3, 1], [3, -3], [8, 3], [-7, 0], [3, 3], [2, 3], [-4, 0], [3, -5], [-12, 2], [8, 2], [-15, -1], [2, 6], [-3, 1], [14, 1], [-9, -3], [-4, -1], [0, 8], [-12, 2], [5, 4], [6, 3], [14, 0], [-1, -8], [-10, 0], [6, -3], [0, -8], [-6, 0], [-4, 2], [-6, 5], [8, 6], [-18, 2], [2, 2], [11, 0], [2, -8], [12, -4], [-4, 10], [12, -4], [9, 1], [-22, 3], [18, 0], [-18, 3], [15, -5], [-3, 1], [17, -6], [-10, 0], [-12, 3], [-7, -2], [-9, 5], [-10, -3], [12, 0], [5, -6], [-6, -5], [5, -8], [-10, 4], [-24, 2], [-1, -10], [-18, -1], [-6, -1], [-1, -6], [8, -8], [-4, -12], [9, 3], [-18, -6], [-4, 4], [-18, -2], [17, -2], [36, 1], [-10, 5], [0, 10], [-7, -4], [-22, 6], [-18, 1], [-9, 5], [5, 2], [6, 10], [2, 11], [-3, 13], [6, -11], [8, 11], [15, 1], [-1, 0], [-15, -9], [24, -2], [-12, 13], [-4, 12], [8, -2], [-28, 0], [3, 9], [3, -1], [6, 14], [-4, -7], [-28, 0], [-6, 0], [30, 1], [12, 6], [5, 0], [-4, 5], [14, 0], [15, 1], [-4, 0], [32, 1], [0, -11], [5, 6], [24, 9], [12, -13], [12, 7], [2, -18], [20, -1], [-15, -1], [6, -5], [26, 4], [-30, 9], [8, 15], [15, 11], [-31, -6], [11, 8], [-19, -4], [-12, 4], [38, 3], [11, 6], [-3, -7], [8, -1], [18, 14], [-7, 6], [-3, -5], [9, 1], [-13, 12], [36, 6], [-16, -8], [15, -7], [-13, 6], [-6, 16], [2, -18], [6, 8], [-1, -16], [-18, -10], [14, 10], [6, -1], [-40, -1], [18, 10], [29, -4], [-24, -4], [-16, 12], [-9, -13], [-4, 2], [-42, -7], [24, 10], [30, 8], [32, -9], [-12, -14], [27, 3], [3, 15], [-1, -24], [-4, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_345_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_345_2_a_i();".
// 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_345_2_a_i(:prec:=2)
    chi := MakeCharacter_345_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(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_345_2_a_i();".
// 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_345_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_345_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-6, 0, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;