
// Make newform 3150.2.m.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_m_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_3150_2_m_a();".
// 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_3150_2_m_a();".
// 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, 0, 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_3150_m();"
function MakeCharacter_3150_m()
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    v := [2, 1, 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_3150_m_Hecke();"
function MakeCharacter_3150_m_Hecke(Kf)
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    char_values := [[-1, 0, 0, 0], [0, 0, 1, 0], [1, 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 := 2;
    raw_aps := [[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, -1, 2, -1], [-4, 0, 4, 0], [2, 0, -2, 2], [0, 2, 0, 2], [2, 4, 2, 0], [2, -2, 0, 2], [6, -3, 0, 3], [-2, -6, -2, 0], [0, 2, -6, 2], [1, 0, -1, 10], [5, 0, -5, -6], [-4, 2, -4, 0], [0, -2, 0, 2], [0, -7, 0, 7], [-7, -2, -7, 0], [0, -2, 4, -2], [0, 0, 0, 10], [0, 4, 0, 4], [2, -4, 2, 0], [-6, 2, 0, -2], [6, -2, 6, 0], [0, 4, 12, 4], [6, 0, -6, -2], [-8, 0, 8, -4], [0, 8, -2, 8], [0, -16, 0, 0], [-2, -8, -2, 0], [0, 6, 8, 6], [12, 0, -12, 0], [0, 4, 12, 4], [-6, 10, 0, -10], [-16, -2, 0, 2], [4, 8, 4, 0], [1, 0, -1, 18], [-9, 0, 9, -2], [-3, -12, -3, 0], [-18, 1, 0, -1], [-12, -5, 0, 5], [0, 4, -20, 4], [-5, 0, 5, 12], [-6, 0, 6, 6], [0, -7, 14, -7], [-4, 0, 0, 0], [4, 0, -4, 16], [4, 0, -4, 4], [0, -3, -4, -3], [12, 4, 12, 0], [-8, -4, 0, 4], [10, -10, 0, 10], [0, -16, -4, -16], [10, 0, -10, 2], [2, 8, 2, 0], [0, -4, 0, 4], [14, -7, 0, 7], [12, -14, 12, 0], [0, -5, -12, -5], [10, 0, -10, -16], [3, 8, 3, 0], [0, 0, 0, 0], [0, -2, -2, -2], [2, 0, -2, 6], [-12, 0, 12, 18], [-4, 16, 0, -16], [-13, 4, -13, 0], [0, 0, 0, 12], [0, -9, -20, -9], [0, 26, 0, 0], [-4, -14, 0, 14], [6, 6, 6, 0], [4, 0, -4, -10], [0, -14, -6, -14], [3, 2, 3, 0], [-10, 14, 0, -14], [8, -10, 8, 0], [0, 19, 0, 19], [0, 0, 10, 0], [-4, 16, 0, -16], [-18, -16, 0, 16], [0, 10, 12, 10], [-14, 0, 14, 6], [0, 7, -6, 7], [-12, 12, -12, 0], [4, 9, 0, -9], [-5, 4, -5, 0], [0, -8, 6, -8], [0, 0, 0, -28], [-14, 0, 14, 0], [-22, 2, 0, -2], [-2, -24, -2, 0], [0, -9, -14, -9], [0, -18, 2, -18], [-1, -22, -1, 0], [24, -4, 0, 4], [0, 26, 6, 26], [14, 0, -14, -16], [2, -22, 0, 22], [-3, 22, -3, 0], [-8, 0, 8, -22], [18, -16, 18, 0], [24, -7, 0, 7], [-28, 12, 0, -12], [-2, 18, -2, 0], [-2, 0, 2, -28], [2, -26, 2, 0], [4, 8, 0, -8], [10, -10, 0, 10], [-4, 16, -4, 0], [2, 0, -2, -10], [0, 0, 0, -34], [0, -10, 32, -10], [-20, 14, 0, -14], [0, -15, 0, -15], [-6, 0, 6, -24], [-29, 0, 29, 6], [10, 6, 10, 0], [14, 9, 0, -9], [16, 17, 0, -17], [-9, 0, 9, -8], [13, 0, -13, -4], [6, 26, 6, 0], [0, 0, 0, 0], [0, 2, 38, 2], [0, 2, 38, 2], [-16, -16, 0, 16], [0, 32, 0, 0], [16, 0, -16, 22], [0, 18, -18, 18], [0, -44, 0, 0], [40, 6, 0, -6], [8, 22, 8, 0], [0, 12, 2, 12], [0, 28, 2, 28], [1, 48, 1, 0], [24, -12, 24, 0], [23, 0, -23, 4], [28, 13, 0, -13], [12, 12, 0, -12], [0, -14, 10, -14], [-20, 0, 20, 12], [-10, 0, 10, 6], [0, -3, 16, -3], [22, 6, 0, -6], [8, 0, -8, -10], [22, 0, -22, 22], [0, 0, -40, 0], [-24, 4, -24, 0], [8, 42, 8, 0], [0, -16, 14, -16], [-19, 0, 19, 26], [-27, 0, 27, 18], [-21, 2, -21, 0], [0, 0, 4, 0], [0, -14, -36, -14], [30, -12, 0, 12], [8, 30, 8, 0], [0, 20, -18, 20], [14, 0, -14, 6], [-32, 6, -32, 0], [8, -40, 8, 0], [0, 0, -36, 0], [16, 0, -16, -24], [25, 22, 25, 0], [-16, -10, 0, 10], [8, -14, 8, 0], [0, -12, 12, -12], [11, 12, 11, 0], [2, 7, 0, -7], [-12, -23, 0, 23], [0, -12, 36, -12], [5, 0, -5, 12], [0, 23, 2, 23], [2, 0, 0, 0], [-18, 2, 0, -2], [0, -30, 2, -30], [-32, 0, 32, -8], [0, 13, 16, 13], [-2, 16, -2, 0], [0, 6, -12, 6], [8, 0, -8, -6], [-22, 0, 22, -22], [-6, 8, -6, 0], [24, 12, 0, -12], [10, -26, 10, 0], [22, 0, -22, -4], [0, 16, 14, 16], [0, -16, 0, -16], [22, 0, -22, -10], [-12, -12, -12, 0], [-50, 6, 0, -6], [0, -12, 8, -12], [-18, 0, 18, 18], [-30, 18, -30, 0], [-20, 28, 0, -28], [14, 0, -14, -26], [2, 0, -2, -42], [-5, -6, -5, 0], [6, 8, 0, -8], [-10, 9, 0, -9], [28, 26, 28, 0], [0, -4, 50, -4], [-24, 16, 0, -16], [5, 0, -5, 24], [0, -5, 34, -5], [-14, -18, -14, 0], [0, -19, 0, 19], [24, -8, 0, 8], [-15, 8, -15, 0], [0, 0, -10, 0], [4, 0, -4, 28], [22, 0, -22, 20], [-30, 2, 0, -2], [34, -12, 0, 12], [26, 24, 26, 0], [0, 10, -26, 10], [-18, 0, 18, 44], [14, -2, 14, 0], [-42, -6, 0, 6], [0, 9, -50, 9], [12, -21, 0, 21], [18, 0, -18, 28], [22, 0, -22, -12], [0, 26, -6, 26], [-6, 22, -6, 0], [-36, 8, 0, -8], [8, 8, 8, 0], [0, 41, -10, 41], [-10, 0, 10, 54], [0, -18, -44, -18], [-36, 24, 0, -24], [0, -3, -8, -3], [-20, 0, 20, 36], [19, 0, -19, -42], [0, -8, 30, -8], [6, 10, 6, 0], [-14, 11, 0, -11], [0, -22, -10, -22], [8, -36, 8, 0], [28, -2, 0, 2], [-4, 0, 4, -40], [0, 18, -34, 18], [0, 2, 0, 0], [8, 24, 0, -24], [0, -40, 0, 0], [0, 2, -40, 2], [0, -40, -4, -40], [-28, -20, -28, 0], [-20, -22, -20, 0], [0, 26, -6, 26], [-12, 0, 12, -20], [0, 10, 42, 10], [-5, 40, -5, 0], [-22, 19, 0, -19], [34, -28, 0, 28], [10, 40, 10, 0], [5, 0, -5, -32], [-2, -14, -2, 0], [12, 0, -12, -16], [-24, 0, 24, 28], [0, 25, 24, 25], [20, 0, -20, 30], [0, 0, 0, -18], [0, 20, 20, 20], [14, -20, 0, 20], [0, -14, -30, -14], [-3, 0, 3, -6], [-10, -26, -10, 0], [12, -19, 0, 19], [23, 26, 23, 0], [-22, 0, 22, 30], [0, -40, 24, -40], [-14, -30, -14, 0], [-42, 0, 42, -6], [-12, 0, 12, -44], [0, 18, -34, 18], [18, 20, 0, -20], [0, -10, 20, -10], [13, 30, 13, 0], [24, 0, 0, 0], [-3, 0, 3, -46], [40, 3, 0, -3], [-22, -4, -22, 0], [0, -30, -36, -30], [-5, 0, 5, -4], [4, 0, -4, 10], [0, 55, -6, 55], [-30, 8, 0, -8], [0, -4, 22, -4], [-4, 0, 4, 0], [12, -4, 12, 0], [0, -30, 36, -30], [-2, 0, 2, 6], [-16, -12, 0, 12], [-54, -17, 0, 17], [23, 12, 23, 0], [-24, 8, 0, -8], [28, -28, 28, 0], [-36, 0, 36, 26], [-38, 0, 38, -18], [0, 12, 52, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_m_a();".
// 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_3150_2_m_a(:prec:=4)
    chi := MakeCharacter_3150_m();
    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_3150_2_m_a();".
// 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_3150_2_m_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 0, 12, 0, 1]>,<13,R![32, 8, 1]>,<17,R![16, -32, 32, -8, 1]>],Snew);
    return Vf;
end function;