
// Make newform 1452.2.i.b in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1452_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1452_i_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_1452_2_i_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_1452_2_i_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 := [1, -1, 1, -1, 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_1452_i();"
function MakeCharacter_1452_i()
    N := 1452;
    order := 5;
    char_gens := [727, 485, 1333];
    v := [5, 5, 3];
    // 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_1452_i_Hecke();"
function MakeCharacter_1452_i_Hecke( : Kf := false)
    N := 1452;
    order := 5;
    char_gens := [727, 485, 1333];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1]];
    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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, 0, 1, 0], [0, -3, 0, 0], [0, 0, 0, 2], [0, 0, 0, 0], [5, -5, 5, -5], [0, -3, 0, 0], [0, 0, 4, 0], [6, 0, 0, 0], [0, 0, 0, -9], [-8, 8, -8, 8], [0, 0, 0, 7], [0, 0, 3, 0], [10, 0, 0, 0], [0, 0, 12, 0], [9, -9, 9, -9], [0, 0, 0, 6], [0, 14, 0, 0], [2, 0, 0, 0], [0, 6, 0, 0], [0, 0, 0, -10], [8, -8, 8, -8], [0, 0, 0, 0], [-3, 0, 0, 0], [1, -1, 1, -1], [18, -18, 18, -18], [0, 0, 0, -2], [0, 0, 6, 0], [-5, 0, 0, 0], [0, 0, -15, 0], [0, -4, 0, 0], [-6, 0, 0, 0], [0, -6, 0, 0], [0, 0, 0, 14], [0, -9, 0, 0], [0, 0, -8, 0], [0, 0, 14, 0], [-14, 14, -14, 14], [18, -18, 18, -18], [0, 0, 18, 0], [0, 0, -24, 0], [0, -5, 0, 0], [0, 0, 0, 12], [0, 11, 0, 0], [-3, 0, 0, 0], [14, 0, 0, 0], [-10, 10, -10, 10], [0, 0, -16, 0], [0, 0, 0, -18], [-5, 5, -5, 5], [-27, 27, -27, 27], [0, 0, -24, 0], [10, 0, 0, 0], [18, -18, 18, -18], [0, 0, 0, 3], [12, 0, 0, 0], [0, 21, 0, 0], [0, 0, 0, 2], [-7, 7, -7, 7], [0, -6, 0, 0], [0, 0, -26, 0], [0, 0, 0, 27], [10, 0, 0, 0], [0, 0, 12, 0], [0, 1, 0, 0], [6, -6, 6, -6], [8, 0, 0, 0], [0, 0, 0, 23], [0, -24, 0, 0], [0, 0, 19, 0], [-15, 0, 0, 0], [0, 0, 0, -6], [0, 0, 0, 10], [22, 0, 0, 0], [0, -2, 0, 0], [-6, 6, -6, 6], [0, 0, 0, 21], [-7, 0, 0, 0], [0, -9, 0, 0], [23, -23, 23, -23], [6, 0, 0, 0], [0, 0, -19, 0], [-18, 18, -18, 18], [0, 0, 0, -11], [4, 0, 0, 0], [0, 0, 30, 0], [15, -15, 15, -15], [0, -1, 0, 0], [9, 0, 0, 0], [-22, 0, 0, 0], [0, -30, 0, 0], [0, 30, 0, 0], [0, 0, 2, 0], [0, 0, 0, 0], [0, 0, 0, 34], [0, 0, -6, 0], [0, 0, -30, 0], [0, 0, 0, -18], [0, 8, 0, 0], [38, -38, 38, -38], [0, 0, -8, 0], [0, 0, 0, 18], [6, -6, 6, -6], [0, 0, -6, 0], [46, 0, 0, 0], [0, -35, 0, 0], [0, 0, 0, 0], [15, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -37], [32, -32, 32, -32], [0, 0, -17, 0], [-3, 0, 0, 0], [0, 0, 26, 0], [0, 0, 0, -38], [0, 0, 45, 0], [0, 16, 0, 0], [-48, 48, -48, 48], [0, 0, 0, 6], [-24, 0, 0, 0], [29, 0, 0, 0], [-10, 10, -10, 10], [0, 3, 0, 0], [-30, 0, 0, 0], [-38, 38, -38, 38], [0, 0, -15, 0], [0, 10, 0, 0], [0, 0, 0, 30], [26, 0, 0, 0], [0, 0, 0, 41], [-16, 16, -16, 16], [0, 30, 0, 0], [0, 0, -10, 0], [19, -19, 19, -19], [-15, 15, -15, 15], [-23, 0, 0, 0], [0, 0, 18, 0], [0, 32, 0, 0], [0, 6, 0, 0], [0, -30, 0, 0], [0, 0, -20, 0], [0, 0, 0, 42], [10, -10, 10, -10], [-6, 6, -6, 6], [0, 0, 0, -41], [0, 0, 24, 0], [0, -31, 0, 0], [-6, 0, 0, 0], [14, 0, 0, 0], [0, -30, 0, 0], [0, 0, 31, 0], [21, 0, 0, 0], [0, 0, 20, 0], [0, 0, 0, 30], [0, 34, 0, 0], [-30, 30, -30, 30], [0, -46, 0, 0], [0, -21, 0, 0], [-13, 13, -13, 13], [0, 39, 0, 0], [-42, 0, 0, 0], [0, 0, 0, -3], [-2, 0, 0, 0], [0, 0, 18, 0], [-21, 21, -21, 21], [0, 0, 0, -12], [-52, 0, 0, 0], [0, 0, 0, 5], [0, 0, -47, 0], [-54, 0, 0, 0], [0, 0, 0, -54], [-38, 38, -38, 38], [0, 0, 0, 0], [1, 0, 0, 0], [0, 40, 0, 0], [0, 0, 0, -9], [0, 8, 0, 0], [0, 9, 0, 0], [0, 0, 0, 2], [53, -53, 53, -53], [64, -64, 64, -64], [-60, 60, -60, 60], [0, 0, 0, 22], [0, 0, -21, 0], [0, 0, 12, 0], [54, -54, 54, -54], [0, 17, 0, 0], [-28, 0, 0, 0], [0, 0, 0, -13], [0, 0, 0, -36], [-38, 38, -38, 38], [0, 0, -36, 0], [0, 16, 0, 0], [0, 0, 0, -3], [-66, 0, 0, 0], [0, -9, 0, 0], [-13, 13, -13, 13], [0, 0, -10, 0], [0, 0, 0, 57], [-12, 12, -12, 12], [0, 0, -66, 0], [-2, 0, 0, 0], [0, 55, 0, 0], [0, -49, 0, 0], [0, 60, 0, 0], [30, 0, 0, 0], [0, 0, -40, 0], [0, 0, 0, -6], [-51, 51, -51, 51], [0, 0, 0, 28], [-11, 0, 0, 0], [0, 0, -21, 0], [0, -68, 0, 0], [-18, 18, -18, 18], [-60, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 32], [0, 0, -42, 0], [0, 0, 18, 0], [-39, 39, -39, 39], [0, -67, 0, 0], [50, -50, 50, -50], [-3, 0, 0, 0], [0, 0, 0, -38], [0, 0, 48, 0], [-17, 0, 0, 0], [0, 0, -51, 0], [60, -60, 60, -60], [0, -28, 0, 0], [12, 0, 0, 0], [5, 0, 0, 0], [0, 0, 0, -16], [0, 0, 4, 0], [0, 0, 0, 9], [-38, 38, -38, 38], [-30, 30, -30, 30], [0, 0, 0, 10], [0, 0, -15, 0], [0, 0, 6, 0], [0, 0, 0, -12], [0, -18, 0, 0], [-52, 52, -52, 52], [0, 0, 20, 0], [55, -55, 55, -55], [-6, 6, -6, 6], [0, 0, -42, 0], [0, -20, 0, 0], [24, -24, 24, -24], [0, -4, 0, 0], [6, 0, 0, 0], [2, -2, 2, -2], [0, -15, 0, 0], [78, 0, 0, 0], [0, 0, -13, 0], [0, 0, 0, 3], [-6, 6, -6, 6], [0, 0, 0, -65], [28, 0, 0, 0], [-39, 39, -39, 39], [0, 0, 0, -58], [-64, 64, -64, 64], [0, -6, 0, 0], [0, 0, 7, 0], [-2, 0, 0, 0], [0, 0, -15, 0], [0, -62, 0, 0], [0, 0, 0, 18], [0, 54, 0, 0], [0, 0, 0, -64], [0, 18, 0, 0], [0, 0, 41, 0], [-20, 20, -20, 20], [0, 0, 0, -71], [-32, 0, 0, 0], [0, 14, 0, 0], [26, 0, 0, 0], [0, -24, 0, 0], [0, 0, 0, -55], [0, 0, -38, 0], [0, 0, 0, 18], [0, 0, -66, 0], [0, -14, 0, 0], [-12, 0, 0, 0], [17, -17, 17, -17], [0, 0, -38, 0], [-24, 0, 0, 0], [0, 0, 50, 0], [0, 0, 0, -42], [-50, 50, -50, 50], [0, 0, 51, 0], [-27, 27, -27, 27], [0, 0, 0, 60], [-9, 0, 0, 0], [0, 48, 0, 0], [0, 0, 10, 0], [30, -30, 30, -30], [0, 0, 0, -26], [0, 0, 0, 57], [-60, 0, 0, 0], [0, 0, 0, 8], [26, -26, 26, -26], [0, -45, 0, 0], [0, 0, 34, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1452_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1452_2_i_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_1452_2_i_b(:prec:=4)
    chi := MakeCharacter_1452_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_1452_2_i_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_1452_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1452_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, 27, 9, 3, 1]>,<7,R![16, -8, 4, -2, 1]>,<13,R![625, -125, 25, -5, 1]>],Snew);
    return Vf;
end function;